THE CALCULUS OF
FINITE DIFFERENCES
THE CALCULUS
OF
FINITE DIFFERENCES
BY
L. M. MILNE-THOMSON, C.B.E.
PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF ARIZONA
EMERITUS PROFESSOR OF MATHEMATICS IN THE ROYAL NAVAL COLLEGE GREENWICH
LONDON,
MACMILLAN & CO LTD
NEW YORK ■ ST MARTIn’s PRESS
1965
This book is copy fight ifi all countries which
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AIORRXSON AND OZBB X,IMITEI>, X.ONDON AND BDINBUROBl
PREFACE
Thk <»r t.liis hook is i-o jirovide a sinijyk' cUid (‘onnootrd aocoimt
of t.h(' sid)joot. of Finite Differences and to present the theory in a
forni which ca,n l)e readily applied.
Two distinct reasons impelled me to undertake this work. Finst,
in ni\' lectures at Greenwich to junior members of the Royal Corps
of Na.val Constructors I have occasion to treat certain aspects of
differ(‘n(U‘ equations ; secondly, the calculation of tables of elliptic
functions and integrals, on which I have been recently engaged,
ga,\u‘ rise to st‘veral int(‘.resting practical difficulties which had to
he overcome. For both these causes my attention has been
(lir(‘et(‘d towards the subject and the lack of a suitable text-book
upon which to draw was brought to my notice. The only com-
prehensive Elnglish treatise, namely Boole’s Finite Difference's,
is long since out of print, and in most respects out of date. My
tirstr idea, was to revise Boole’s book, but on looking into the matter
it appeared that such a course would be unsatisfactory, if not
impracticable. 1 therefore decided to write a completelj^ new
work ill which not only the useful material of Boole should find a
place, but in which room should also be found for the more modern
developments of the finite calculus.
My aim throughout has been to keep in mind the needs of the
beginner, so that the book may be regarded as suitable for a first
course as well as for more advanced reading. I do not, however,
believe that the needs of the beginner in a mathematical subject
a, re best served by eschewing all but the most elementary math(‘.-
matical apparatus. Rather, his interest in the subject may well
form an adequati‘. o])portunity for enlarging liis outlook on the
science of ruatheniatics, so that he may the better be enabled to
distinguish and appreciate' the connection of the whole system
a,nd the relative dependency of its several parts. Consequently
1 have not hesitated to use the mathematical process or terminology
which has appeared to me most appropriate to the immediate
object in view. On the other hand whenever this course seems to
PREFACE
vi
lead beyond tbe elementary matters, with which all who embark
on the reading of a mathematical book must be presumed to be
acquainted, I have included the necessary definitions or proofs
as part of the text or have given accessible references to treatises
which can ordinarily be found in any mathematical library. In
this way it has been possible to treat the subject in a simple yet
rigorous manner.
The subject-matter falls naturally into two main divisions which
may be subsumed under the headings Interpolation and Difference
Equations. The pioneer in interpolation was undoubtedly Briggs,
whose work was largely of an arithmetical character. Newton
was the originator of the systematic theory and his divided difference
formula is really the fundamental basis of all the usual metliods
of polynomial interpolation. Gregory was probably an independent
discoverer in the same field.*
The present work therefore starts with divided differences in
Chapter I ; and in a general sense Chapters III, IV and VII may
be regarded as elaborations of Newton’s work. Chapter V on
reciprocal differences, describes a method of interpolation, due to
Thiele, by means of rational functions, which is more general
than polynomial interpolation, and which will possibly be new to
many Enghsh readers. Chapter VI introduces the generalisations,
due to Norlund, of Bernoulli’s polynomials, but here they are
treated by a symbohe method, which seemed to me to be as effi-
cacious and in many ways more suitable than Norland’s method,
which is founded upon a different principle. By means of these
generalisations the subject of numerical differentiation and integra-
tion assumes a unified aspect which hardly seems to be attainable
without them. Chapters I to VII therefore form a suitable intro-
ductory course and will make very little demand on the reader’s
previous mathematical knowledge. I have tried to meet the
requirements of those who wish to make numerical applications
by giving the formulae in a manner suited to direct use with a
table of data. The numerical illustrations scattered through these
chapters are mainly of a simple kind which can be easily worked,
for it is not my purpose to obscure principles by unnecessary arith-
metic. The subject-matter of some of these examples is perhaps
* See H. W. TumbuU, p. 101, footnote.
PREFACE
vii
of an unusual nature, but this is intentional in order to lend variety
to the applications. In the chapters on Interpolation I have followed
Steffensen’s excellent example in lapng much stress on the remainder
term, which measures the error committed in using an interpo-
lation formula. Indeed, no formula has been given which is
unaccompanied by a means of estimating the remainder.
The part of the book which deals with difference equations
begins with Chapter VIII, which expounds Norlund’s method of
treating the summation problem. Chapter IX applies these
methods to elaborating the theory of the Gamma function. In
Chapter X, I have attempted to give a consecutive account of the
salient properties of factorial series, which, I hope, wiU prove
interesting in itself. The object of this chapter is to develop the
properties of the series in which the solutions of difference equations
find their natural expression. Chapter XI discusses the difference
equation of the first order ; the hnear case is completely elucidated,
and certain amenable non-linear forms are treated. This chapter
includes an investigation of the exact difference equation of the
first order. The methods of this, and of succeeding chapters,
are illustrated by simple worked examples in the text. Chapter XII
considers the properties of the general linear equation, including
the application of generalised continued fractions treated by matrix
methods. Chapter XIII deals with the important case of constant
coefficients. Here the theory is complete, in the sense that the
solution can be explicitly obtained. I have dealt with this equation
at some length both by Boole’s method and by a method of my
own, which seems well adapted to applications of a geometrical
or physical nature, and which is analogous to Heaviside’s method
for differential equations. The linear equation with constant
coefficients has recently come into prominence in connection with
various physical and mechanical problems ; for example in the
theory of Structures. Chapters XIV and XVI develop the solution
of linear difference equations with variable coefficients by means
of Boole’s operators, which I have generalised in order to render
the treatment more complete. Chapter XV gives an alternative
treatment founded on Norland’s use of Laplace’s transformation.
Chapter XVII gives two fundamental theorems of Poincar4 and
Perron on the asymptotic properties of the solutions of a certain
type of linear difference equation. The proof of Perron’s theorem
PREFACE
viii
is made to depend upon the properties of a certain class of simul-
taneous linear equations in infinitely many unknowns. The theory
is so interesting and so closely connected with finite differences
that it has seemed worth while to give Perron’s treatment in extenso.
Operational and symbolic methods have been freely used through-
out the book, and it is hoped that the manner of presentation here
given will be found free from the objections often associated with
their use. Indeed it has always seemed to me that symbolic
methods constitute the essence of the finite calculus. My choice
of notations has therefore been made with a view to facilitating
the statement and application of operational methods, and to
stressing the analogies with the infinitesimal calculus.
In stating theorems I have as far as possible associated the name
of the discoverer as sufficient indication of the origin, but it must
not be assumed that the method of presentation is in every case
that in which the theorem was originally given. Indeed in the
case of the work of the older analysts it would be easy, but un-
profitable, to point out defects and lack of rigour in many of their
proofs.
My labour in correcting the proof sheets has been greatly lightened
by Professor H. W. Turnbull, F.R.S., who has read the first proof
and made many valuable suggestions both mathematical and
historical ; and by Dr. A. G. Aitken, F.R.S.E., who has performed
the same kindly office, has supplied many original examples, and
has verified the numerical work. To both these friends I wish to
express my lively thanks for assistance which has helped me to
remove many imperfections both of expression and demonstration.
For any blemishes which may remain I am solely responsible, but
I am led to express the hope that the work will be found to be free
from important errors. I take this opportunity of expressing my
thanks to the officials of the Glasgow University Press for the
ready way in which they have met my somewhat exacting require-
ments.
L. M. MILNE-THOMSON.
Mathematics Department,
Royal Naval Colleoe,
Greenwich,
Jvlif 1933.
CONTENTS
PAGE
Introduction xxi
Notations xxii
CHAPTER I
DIVIDED DIFFERENCES
1*0. Definitions 1
M . Newton’s interpolation formula with divided differences - - 2
M5. Rolle’s Theorem 4
1*2. Remainder term in Newton’s formula 5
1*3. Divided differences are symmetric functions of the arguments 7
1*31. Divided differences of 7
1-4. Lagrange’s interpolation formula 8
1-5. Expression of divided differences by means of determinants • 9
1- 6. Divided differences expressed by definite integrals - - - 10
1*7. Divided differences expressed by contour integrals - - - 11
1*8. Divided differences with repeated arguments - - - - 12
1*9. Interpolation polynomials 14
Examples I 17
CHAPTER II
DIFFERENCE OPERATORS
2*0. Difference notation 20
2- OL Central difference notation . , - - 22
2*1. Difference quotients 23
2*105. Partial difference quotients - - - - 24
2*11. Difference quotients of factorial expressions - 25
2*12. Expansion of a polynomial in factorials - - 27
2*13. Successive difference quotients of a polynomial 28
2*14, Difference quotients of - - - - 29
2*2. Properties of the operator A - - - - 30
u»
2*3. The operator y 31
(M OJ
X
CONTENTS
24. The operator E“
241. Herschel’s Theorem
242.
243. <55>(E“)a®w(.'r)=a«=</)(a"E“)?^(^) - - - •
2*5, Relations between A, E^s
ta
2*51. The analogue of Leibniz’ Theorem - - - -
2*52. Difference quotients of
2*53. Difference quotients of zero
2*54. Difference quotients in terms of derivates
2*6. The summation operator p”^
2*61. A theorem on the value of -
2*62. Relation between sums and functional values -
2*63. Moments
2*64. Partial summation
2*7. Summation of finite series
2*71. Summation of factorial expressions of the form xW
•72. Polynomials - - -
•73. Factorial expressions of the form -
2‘74. A certain type of rational function - . - .
2*75. The form a® a polynomial
2*76. The form (re), <^(x) a polynomial
2*77. Unclassified forms
Examples II
CHAPTER III
INTERPOLATION
3*0. Divided differences for equidistant arguments -
3*1. Newton’s interpolation formula (forward differences)
3*11. Nevijon’s interpolation formula (backward differences)
3*12. The remainder term
3*2. Interpolation formulae of Gauss - . . .
3*3. Stirling’s interpolation formula - - . _
3*4. Bessel’s interpolation formula - - - - _
3*41. Modified Bessel’s formula - - - - .
3*5. Everett’s interpolation formula - -
3*6. Steffensen’s interpolation formula - - _ .
3*7. Interpolation without differences - - - _
3*81. Aitken’s linear process of interpolation by iteration
3*82. Aitken’s quadratic process
3*83. Neville’s process of iteration
Examples III
PAGE
31
32
32
32
33
34
35
36
37
37
39
39
40
41
42
42
43
44
45
46
46
48
49
56
57
59
61
63
67
68
71
72
74
75
76
78
81
84
lO lO
CONTENTS
CHAPTEE IV
NUMERICAL APPLICATIONS OF DIFFERENCES
PAGE
4*0. Diilerences when the interval is subdivided . - . • 87
4*1. Differences of a numerical table 88
4*2. Subtabulation 91
4*3. Inverse interpolation 95
4*4. Inverse interpolation by divided differences - . - ■ 96
4*5. Inverse interpolation by iterated linear interpolation - 97
4*6. Inverse interpolation by successive approximation - - • 99
4*7. Inverse interpolation by reversal of series - - - -100
Examples IV 101
CHAPTER V
RECIPROCAL DIFFERENCES
5*1. Definition of reciprocal differences - - - 104
5*2. Thiele’s interpolation formula - - - - 106
5*3. Matrix notation for continued fractions - - 108
5*4. Reciprocal differences expressed by determinants 110
5*5. Reciprocal differences of a quotient - - - 112
5*6. Some properties of reciprocal differences - 114
5*7. The remainder in Thiele’s formula - - - 116
•8. Reciprocal derivates ; the confluent case - - 117
•9. Thiele’s Theorem 119
Examples V 122
CHAPTER VI
THE POLYNOMIALS OF BERNOULLI AND EULER
6*0. The <f) polynomials 124
6*01. The (3 polynomials 126
6*1 Definition of Bernoulli’s polynomials 126
6*11. Fundamental properties of Bernoulli’s polynomials - - - 127
6*2. Complementary argument theorem 128
6*3. Relation between polynomials of successive orders - - - 129
6*4. Relation of Bernoulli’s polynomials to factorials - - - 129
6*401. The integral of the factorial 131
6*41. Expansion of xW in powers of a: 133
6*42. Expansion of x'' in factorials 133
6*43. Generating functions of Bernoulli’s numbers - - - - 134
CONTJi:Nl’fc5
xii
PAGE
6*5. Bernoulli’s polynomials of tlic first ord(‘r - 136
6-501. Sum of the vtii powers of the first n integers 137
6-51. Bernoulli’s numbers of the first order - - 137
6-511. Euler-Haclaurin Theorem for polynomials - 139
6-52. Multiplication Theorem 141
6-53. Bernoulli’s polynomials in the interval (0, 1) - 141
6-6. The 7] polynomials 142
6-7. Definition of Euler’s polynomials . - - 143
6-71. Fundamental j^roperties of Euler’s polynomials 144
6*72. Complementary argument theorem . ~ 145
6-73. Euler’s polynomials of successive orders - 145
6-8. Euler’s polynomials of the first order - - 146
6-81. Euler’s numbers of the first order - - - 147
6-82. Boole’s Theorem for polynomials . - - 149
Examples \T 150
CHAPTER VII
NUMERICAL DIFFERENTIATION AND INTEGRATION
7-0. The first order derivate 154
7*01. Derivates of higher order . - . . 155
7-02. Markoff’s formula 157
7*03. Derivates from Stirling’s formula - - - 159
7-04. Derivates from Bessel’s formula ... 161
7-05. Differences in terms of derivates ... 162
7-1. Numerical integration 162
7-101. Mean Value Theorem 163
7*11. Integration by Lagrange’s interpolation formula 164
7*12. Equidistant arguments 165
7*13. Remainder term, n odd 166
7-14. Remainder term, even - . - - - 167
7*2. Cotes’ formulae 168
7-21. Trapezoidal rule 170
7*22. Simpson’s rule 171
7*23. Formulae of G. F. Hardy and AVcddle ... 171
7*3. Quadrature formulae of the open tyq:)e ... 172
7*31. Method of Gauss 172
7*33. Method of Tschebyscheff . . . _ . 177
7*4. Quadrature formulae involving differences - . 180
7*41. Laplace’s formula 181
7*42. Laplace’s formula applied to differential equations ■ 183
7*43. Central difference formulae - - - . . 184
CONTENTS xiii
PAQK
7’5. Euler- Maclaurin formula - 1^7
7*51. Apx^lioation to finite summation 191
7-6. Gregory’s formula - - - 191
7-7. Summation formula of Lubbock 193
Examples VII - - - 196
CHAPTER VIII
THE SroiMATION PROBLEM
8*0. Definition of the principal solution or sum .... 201
8*1. Properties of the sum 204
8T1. Sum of a polynomial 208
8*12. Repeated summation 208
8*15. Proof of the existence of the principal solution (real variable) - 209
8*16. Bernoulli’s polynomials iL (a* I CO ) 213
8*2. Differentiation of the sum 213
8*21. Asymptotic behaviour of the sum for large values of a - - 214
8*22. Asymptotic behaviour of the sum for small v^alues of co - - 216
8*3. Fourier series for the sum 218
8*4. Complex variable. Notation. Residue Theorem - - - 220
8*41 . Application of Cauchy’s residue theorem .... 222
8*5. Extension of the theory 226
8*53. The sum of the exponential function - - - - - 231
8*6. Functions with only one singular point 232
8*7. An expression for F{x* I - CO ) 238
Examples VUI 238
CHAPTER IX
THE PSI FUNCTION AND THE GAMMA FUNCTION
9*0. The function 'T' (a* j co) 241
9*01. Differentiation of the Psi function 241
9*03. Partial and repeated summation 243
9*1. Asymptotic behaviour for large vahic‘S of a* . . . . 244
9*11. Partial fraction development of 'P (a: 1 CO ) _ . . . 245
9*2. Multiplication theorem for the Psi function . - . - 246
9*22. Fourier series for "P (a;) 247
9*3. Gauss’ integral for 'P (a*) 247
9*32. Poisson’s integral 248
9*4. Complementary argument theorem for the Psi function - - 249
9*5. The Gamma function 249
xiv
CONTEISTTS
PAGE
9*52.
Schlomilch’s infinite product for r(a; + 1)
250
9*53.
Infinite products expressed by means of F (a;)
251
9-54.
Complementary argument theorem for F (a:) -
251
9-55.
The residues of F (a:) -
262
9-56.
Determination of the constant c -
262
9-6.
Stirling’s series for log F (a; +
253
9-61.
An important limit - . . . -
254
9-66.
Generalised Gamma function F(a: | co) -
255
9-67.
Some definite integrals
266
9-68.
Multiplication theorem of the Gamma function
257
9-7.
Euler’s integral for F (a;)
257
9-72.
Complementary Gamma function Fi(a;) -
258
9-8.
H3q)ergeometric series and function F {a, b ; c ; x)
260
9-82.
H3q)ergeometrie function when a; = 1
261
9-84.
The Beta function B{x, y) -
262
9-86.
Definite integral for the hypergeometric function
264
9-88.
Single loop integral for the Beta function
265
9-89.
Double loop integral for the Beta function
266
Exasiples IX
267
CHAPTER X
EACTORIAL SERIES
10*0. Associated factorial series . 272
10-02. Convergence of factorial series 273
10-04. Region of convergence 275
10*06. Region of absolute convergence ------ 276
10-07. Abel’s identities 276
10-08. The upper limit of a sequence 277
10-09. Abscissa of convergence. Landau’s Theorem - - - 279
10-091. Majorant inverse factorial series 283
10-1. Series of inverse factorials 284
10-11. Uniform convergence of inverse factorial series - - - 284
10-13. The poles of (rr) 287
10*15. Theorem of unique development 288
10-2. Application of Laplace’s integral ; generating function - » 288
10*22. Order of singularity and the convergence abscissa - - - 292
10-3. The transformation (ar, a: +m) 293
10*32. The transformation {x, xjcxi) 294
10*4. Addition and multiplication of inverse factorial series - - 295
10*42. Differentiation of inverse factorial series - - - - 297
10*43. An asymptotic formula 298
CONTENTS
PAGE
10*44. Integration of inverse factorial series - - 299
10*5. Finite difference and sum of factorial series - 300
10*6. Newton’s factorial series - - - - 302
10*61. Uniform convergence of Newton’s series - - 302
10*63. Null series 304
10*64. Unique development 305
10*65. Expansion in Newton’s series ; reduced series - 306
10*67. Abscissa of convergence of Newton’s series - 309
10*7. Majorant properties 310
10*8. Euler’s transformation of series . - . 311
10*82. Generating function 312
10*83. Laplace’s integral and Newton’s series - - 314
10*85. Expansion of the Psi function in Newton’s series 315
10*9. Application to the hypergeometric function - 316
Examples X 317
CHAPTER XI
THE DIFFERENCE EQUATION OF THE FIRST ORDER
11*0. Genesis of difference equations 322
11*01. The linear difference equation of the first order - - - 324
11*1. The homogeneous linear equation 324
1 1 *2- Solution by means of the Gamma function. Itational coeflicients 327
11*3. Complete linear equation of the first order . . - . 328
11*31. The case of constant coefficients 329
11*32. Application of ascending continued fractions - - - . 330
11*33. Incomplete Gamma functions 331
11*34. Application of Prym’s functions 332
11*4. The exact difference equation of the first order - - - 334
11*41. Multipliers 339
11*42. Multipliers independent of a: 339
11*43. Multipliers independent oi u 340
11*5. Independent variable absent. Haldane’s method - - - 341
11*51. Boole’s iterative method 343
11*6. Solution by differencing. Clairaut’s form . _ . . 344
11*7. Equations homogeneous in w 346
11*8. Riccati’s form 346
11*9. Miscellaneous forms 347
Examples XI 348
XVI
CONTENTS
CHAPTEE XII
GEXEEAL PEOPEETIES OF THE LINEAE BIFFEEENCE
EQUATION
PAGE
12*0. The homogeneous linear difference equation - - - - 351
12*01. Existence of solutions 352
12*1. Fundamental system of solutions 353
12*11. Casoratfs Theorem 354
12*12. Heymann’s Theorem 357
12*14. Relations between two fundamental systems - . . . 359
12*16. A criterion for Hnear independence 360
12*2. Symbolic highest common factor 361
12*22. Symbolic lowest common multiple 363
12*24. Reducible equations 366
12*3. Reduction of order wFen a solution is known - - - - 367
12*4. Functional derivates 369
12*5. Multiple solutions of a difference equation - - - . 379
12*6. Multipliers. Adjoint equation 372
12*7. The complete linear equation. Variation of parameters - - 374
12*72. Polynomial coefficients 377
12*8. Solution by means of continued fractions - . . - 373
Examples XII 381
CHAPTER XIII
THE LINEAR DIFFERENCE EQUATION WITH CONSTANT
COEFFICIENTS
13*0. Homogeneous equations 3g4
13*02. Boole’s symbolic method 387
13*1. Complete equation 333
13*2. Boole’s operational method 392
13*21. Case I, <j){x) = x'^ . 39^
13*22. Case II, <j:i{x) = - - - . . . , _ _ gqg
13*23. Casein, (^{x) = a^M{x) 393
13*24. The general case
13*25. Broggi’s method for the particular solution - - - - 401
13*26. Solution by undetennined coefficients 493
13*3. Particular solution by contour integrals - - - - - 494
13*32. Laplace’s integral
13*4. Equations reducible to equations with constant coefficients - 408
CONTENTS xvii
PAGE
13*5. Milne-Thomson’s operational method 410
13*51. Operations on unity 411
13*52. Operations on a given function X 412
13*53. Application to linear equations with constant coefficients - 413
13*54. Simultaneous equations 415
13*55. Applications of the method 415
13*6. Simultaneous equations 420
13*7. Sylvester’s non-linear equations 420
13*8. Partial difference equations with constant coefficients - - 423
13*81. Alternative method 425
13*82. Equations resolvable into first order equations - - - 426
13*83. Laplace’s method 427
Examples XIII 429
CHAPTER XIV
THE LINEAR DIFFERENCE EQUATION WITH RATIONAL
COEFFICIENTS. OPERATIONAL METHODS
14*0. The operator p 434
14*01. The operator tt 436
14*02. Inverse operations with tt ----- - 437
14*03. The operators tti, pi 439
14*1. Theorem I 439
14*11. Theorem II 440
14*12. Theorem HI 440
14*13. Theorem IV 442
14*14. Theorem V 443
14*2. Formal solution in series 445
14*21. Solution in Newton’s series 448
14*22. Exceptional cases 451
14*3. Asymptotic forms of the solutions 457
14*31. Solutions convergent in a half-plane on the left - - 459
14*4. The complete equation 460
14*5. Monomial difference equations 461
14*6. Binomial equations 465
14*7. Transformation of equations. Theorems VI, VII, VIII - 467
14*71. Equation with linear coefficients 469
14*73. The equa^tion {ax^ -{-hx+c)u{x) +{ejc +f)ii{X’-l) -{■gu{x -2) = 0 472
14*75. The equation {ax^ +hx +c)Au-\- {ex +f)Au+gu‘- 0 474
14*8. Bronwin’s method 475
14*9. Linear partial difference equations - - 475
CONTENTS
PAGE
476
477
xviii
14-91. Laplace’s method for partial equations
Examples XIV . - - -
CHAPTER XV
THE LINEAR DIFFERENCE EQUATION WITH RATIONAL
COEFFICIENTS. LAPLACE’S TRANSFORMATION
15*0. Laplace’s transformation - - - 478
15*1. Canonical systems of solutions - - - 482
15*2. Factorial series for the canonical solutions 485
15*3. Asymptotic properties - - - . 487
15-31. Casorati’s determinant . , » - 488
15*4. Partial fraction series . . . . 490
15-5. Laplace’s difference equation - - - 491
15-51. Reducible cases 493
15-52. Hypergeometric solutions - - - 494
15-53. Partial fraction series - . . . 495
15-54. Relations between the canonical systems - 496
15-55. The case aj =^2 498
15-6. Equations not of normal form - - - 500
Examples XV 501
CHAPTER XVI
EQUATIONS WHOSE COEFFICIENTS ARE EXPRESSIBLE
BY FACTORIAL SERIES
16-0. Theorem IX 504
16-01. Theorem X 505
16-1. First normal form 5O8
16-2. Operational solution of an equation of the first normal form - 509
16-3. Convergence of the formal solution 511
16-4. Example of solution 510
16-5. Second normal form 5I8
16-6. Note on the normal forms - - - . . . - 519
Examples XVI
CHAPTER XVII
THE THEOREMS OP POINCARE AND PERRON
1 / *0. Tlie linear ei^uation witli constant coefficients - 523
17-1. Poincare’s Theorem
CONTENTS
xix
PAGE
17-2. Continued fraction solution of the second order equation - - 532
17-3. Sum equations - 534
17*4. Homogeneous sum equations with constant coefficients.
Theorem I 537
17*5. A second transformation of sum equations . . - . 539
17*6. General solution of sum equations. Theorem II - - - 542
17*7. Dilference equations of Poincare’s type. Perron’s Theorerq - 548
Examples XVII 550
Index 553
INTRODUCTION
Let /(a:) be a given function of the variable x. The Differential
Calculus is concerned with the properties of
f{x) - Dm = lim
W-^0 ^
which is still a function of the single variable x. On the other hand
the Calculus of Differences is concerned with the properties of
w a)
which is a function of the two variables x and co.
More generally, in contrast with the Infinitesimal Calculus, the
Finite Calculus is concerned with the values of a function at a set of
isolated points and with such properties as may be derivable there-
from.
Suppose then that we are given the numbers/(:r^),/(a^2)?/(%)5 ••• ?
and an argument x different from 0:3, ... . Among the subjects
of enquiry which naturally present themselves are the following.
(i) The determination of /(a;) from the given functional values.
This is the Interpolation problem.
(ii) The determination of
^ a
These are the problems of Numerical Differentiation and Integration.
Extending our enquiries in another direction we are led to con-
sider the properties of the functions J[x) defined by the equation
CO
where g{x) is a given function. This constitutes the Summation
problem, which is analogous to the problem of integration in the
Integral Calculus.
xxii INTKODUCTION
On the theory of summation we are able to found in a satisfactor}^
manner the theory of the Gamma function which plays such an
important part in the Calculus of Differences.
Consideration of more general relations between/(a?), /(a? -f co), . . . ,
/(x-f noj) brings us to the study of difference equations, which are
analogous to the differential equations of the Infinitesimal Calculus.
NOTATIONS
The following list, which is intended only for reference, contains
the s}Tnbols, operators, and functions which occur most frequently
in this book. The numbers refer to the sections where explanations
are given.
Symbols
'm\ __ m(m-l) ... {m-7i + l)
nJ ~ n!
lim /(a*), the limit of f(x) when
X tends to a,
lim sup a;„. Upper limit, 10*08.
n-^x
fix) ~ g{x), 8-22.
==, symbolic equivalence, 2*4:1.
R{x), the real part of x, 8*4.
Rn{^), Remainder Terms,
M, 7*11.
|x|, arg X, 8*4.
C7, Arbitrary Periodic Punction,
IM.
... Divided Difference,
1*0.
= x(x-o)){x-2(x>) ...{x-moi + oi),
2*11.
~ (r+co)-i(a;H-2co)~h..
(r+mco)~\ 2*11.
n
A Difference Quotient of
Zero, 2-53.
Bernoulli's Numbers of
order n, 6T.
Euler's Numbers of order n,
6*7.
Bv, Bernoulli's Numbers, 6*5.
Ey, Euler's Numbers, 6-8.
OK), 9*8.
NOTATIONS
Operators
Difference Operator, 2-0.
S, [jlS, Central Difference Oper-
ators, 2*01.
Difference Quotient Oper-
" ator, 2*1.
D, Differentiation Operator, 2*1.
Partial Difference Quotient,
“ 2-105.
V, 2*3.
E“, 2-4,
P~^, Summation Operator, 2-6.
p, p„, Eeciprocal Difference, 5-1.
T, Reciprocal Derivate, 5-8.
Sum Operator, 8-0.
Tc, p, 14-01, 14-0.
TTi, pp 14-03.
Functions
exp X, Exponential Function.
Bernoulli's Polynomial
of order n, 6-1.
B^{x\ co). Generalised Bernoulli’s
Polynomial, 8-16.
Euler’s Polynomial of
order n, 6-7,
By{x), Bernoulli’s Polynomial,
6-5.
E^{x), Euler’s Polynomial, 6-8.
Periodic Bernoulli Func-
tion, 7-5.
B(ir, y), Beta Function, 9*84.
r(a::). Gamma Function, 9*5.
Viix), Complementary Gamma
Function, 9-72.
r(x|6)), Generalised Gamma
Function, 9-66.
co), Psi Function, 9-0.
'F(x), Psi Function, 9-0.
F{x \ co). Sum Function, 8-0.
F(a,b;c;x), Hypergeometric
Function, 9-8.
Q(a;), Factorial Series Sum
Function, 10-1.
CHAPTER I
DIVIDED DIFFERENCES
1-0. Definitions. Consider a function /(a;) whose values are
given for the values of the variable x. These latter
values we suppose to be all different.
The divided difference oif[x) for the arguments Xq, x-^ is denoted
by and is defined by the relation
1 /(^o) fi^i) _„/(%)“/(^o) _ T
Xi-Xn
Sinularly we define the divided difierence of arguments 2:25 ag by
[=<=1^2] =
and so on.
Two divided differences of two arguments having a common
argument can be used to define a divided difference of three argu-
ments. Thus the divided difference [xqX-^x,^ of the three arguments
^0, x-^, is defined by
[XqX-^x^ = [^0^1] ~ [^1^2] _ [^o%] ~ [^2^1]
^0 ~ ^2 ^0 ““ ^2
Proceeding in this way we can form divided differences of ti+l
arguments when we have defined the divided differences of n argu-
ments. Thus
\x X r 1 — “* "" ••• ^n]
Jy(\ —
* Other notations are ^"(vi ... x„), d”f{x), x^ x„). It wiU be
proved in 1*3 that divided differences are symmetric functions of their
arguments so that the order of symbols within the bracket is immaterial.
[1-0
DIVIDED DIEEERENCES
These results may be exhibited in, a scheme of divided differences as
follows :
Xq
fi^a)
[XqX^]
[^0%^2]
1 — 1
1
X2
f3)
1 — 1
1
C9
1 1
{x^x^x^
[XjX^^i]
/fe)
[^3^4]
1 1
■«#
4
^4
•
1 — 1
/w
1-1. Newton’s Interpolation Formula with
Differences. Writing % for we have by definition
. . . a:„] = - ^
^ 2 X-Xn 'T.-rr.
X-Xn
X-X,
n-l
Divided
[xxi ... cr„_J = - .
[ afi— B-aJ x-x„_s ‘ x-x„_2
[xx-
^ *•* x-x, X — Xi’
Mm XJ //• _ O* 'T* _
o/ ~ JU'Y Jj
By repeatedly substituting for the second member on the right of
each identity its value as given by the succeeding identity, we have
[Zia^ — iCn] [x^Xz — x^-^
x-x^ {x-x„){x-Xn_-^
[XjX^ ... x„_^ [x^
[ax5ia:2...a:„] = -^
{x-x„){x-x„_^{x-x„_^) (x-x„){x-x„_j)...(x-xzj
M) /(^)
(X-X^ ... {X-X2)(x-Xi) '^{X-X^) ... (X-Xj)
M]
or
DIVIDED DIFPEKENCES
f{x) = f (xi) + (a; - Xj) [x^x^ +{x- aj^) (a: - x^) [x^x^x^] +...
+ (a; - aji) {x-x^)...(x- x,_j} [x^x^, . . . a;J + . . .
+ (a; - a^i) (a: - aja) . . . (aj - a;„_i) [x^x^ ■ • • a3„]
+ (a: - aji) {x-x^...{x- x„) [xx^x^ • • • a;„],
or
n- 1
(1) f{x) =f{x-i)+'^{x-Xi){x-Xi) ... {x-x,)[x^x^ ... a;,J + .R„(a;),
S-1
(2) wliere jR„ (a;) = (x - Xj) (x - Xg) . . . (x - x„) [xx^Xg . . . x„].
This is Newton’s general interpolation formula with the remainder
term i2n(x). The formula is of course a pure identity and is therefore
true without any restriction on the form of /(x).
By means of this formula the evaluation of a function /(x) whose
value is known for the values x^, Xg, , x^ of the variable is reduced
to the problem of evaluating the remainder term (x). Should this
term be known or negligible, the required value/(x) can be calculated
from Newton’s formula.
It should be observed that no particular rule is laid down for the
sequence of the arguments x, Xj, Xg, ... , x^, which need not be in
ascending or descending order of magnitude.
If /(x) be a polynomial of degree n-1 in x, then
[KKi] =
/W-/K)
x-x^
is of degree - 2 in x. Hence the operation of taking the divided
difference of a polynomial lowers the degree by unity. Conse-
quently the divided difference of the (n- l)th order of a polynomial
of degree n-1 is constant, and therefore the divided difference of
the nth order is zero, that is, [xx^ ... x„] = 0.
In this case i2„(x) = 0, so that the value of f(x) given by the
formula
n— 1
(3) f{x)= f{Xj) +'^{x-xj)...{x~x,) . . . x,+^}
is exact.
If/(x) be not a polynomial, we see that
Rn{x) = 0, for X = X^, Xg, ... , Xn,
DIVIDED DIFFERENCES [i-1
SO that the right-hand member of (3) yields the polynomial of degree
n~l whose value coincides with the value of /(cc) for
X = Xj, Xo, ... , Xn-
Example. Find approximately the real root of the equation
^3 _ 2y -5 = 0.
Let X = ]f-2y-6.
This relation defines a function y=f(x). We want the value of
/(O). Attributing suitable values to we obtain the following
ta])le of divided differences.
X
m
- 1-941
1-9
+ -10627
- 1-000
2-0
+•09425
- -0060
+ -0005
+ 0-061
2-1
+ -08425
-•0044
+•0003
+ 1-248
2-2
+ -07582
- -0034
+ 2-567
2-3
Thus approximately, using (3) above, w^e have
2/ = 2-0 -t- 1 X *09425 + 1 x *061 x *0044 + 1 x *061 x 1*248 x *0003
= 2*09454.
The corresponding value of x is -0*00013 and the above value of
y is in error by about one unit in the last digit.
1 *16. Rollers Theorem. In order to discuss the form of the
remainder term in hiewton’s formula we need the following theorem
known as Eolle’s theorem.
If the functioyi f{x) be continuous and differentiable in the interval
a:^x:^b, where a, h are two roots of the equation f{x) = 0, then the
equation f^ {x) = 0 has at least one root interior to the interval (a, b).
Proof If f{x) have the constant value zero, the theorem is
evident. If/(a;) be not constantly zero, it will take positive values
or negative values. Suppose that f{x) takes positive values. Then
DIVIDED DIFFERENCES
M5]
f{x) being continuous will attain a maximum value M for some
point 5 such, that a <^<h. Thus, if h be positive,
M+h)-M)
h
is negative or zero and hence the limit when A-;-0, namely
cannot be a positive number, that is,
Similarly, by considering the ratio
we prove that
Thus we have /'(5) = 0 and the theorem is proved. In the
same way we prove the theorem for the case when f{x) takes
negative values.
1 *2. The Remainder Term. From M (2), we have
Rni^) =f{x)-Pn-i{x)
n — 1
where Pn-M = ...{x-x,)[x.^x^...x,^^l
so that is a polynomial of degree n-1, and its (w-l)th
derivate is
(1) Ptl^\x)^{n-l)nx,x^...x^].
Hitherto / {x) has been unrestricted. We now suppose that in the
interval {a, b) bounded by the greatest and least of x, x^, x^, ...,x„
the function /(i) of the real variable t, and its first n—1 derivates
are finite and continuous and that/(")(«) exists.
Then since R„{t) vanishes when t = 3^,x^,..., x„, by Rolle’s
theorem it follows that X(t) vanishes at n--l points of (a, b) and
therefore by a second application of the theorem that E'n {t) vanishes
at n-2 points of {a, h). Proceeding in this way we see that
= 0 where 7] is some point of (a, 6).
Thus =
0 DIVIDED DIFFERENCES 1^='^
SO that
(2) [^1^2 ••• ^n] = - rjr ’
wliich is a formula expressing the divided difference of order n-1
in terms of the (w-l)th derivate of f{x) at some point of {a, b).
Hence we have
[xXjX^ ...X„] = ,
where ^ is some point of (a, b). We have therefore
(3) B„{x) = {x-Xt){x-x^ ... {x-x„)
where ^ is some point of {a, b).
This important result enables ns to find an upper limit to the
error committed in omitting the remainder term, provided that we
can find an upper limit for the ^th derivate of/(0 in the interval
bounded by the greatest and least of x, x^j x^^ ••• >
Exam'ple. Find an approximate value of logio 4*01 from the
following table :
••• a-nj — ’
X
logic ^
4-0002
0-6020 817
+•108431
4-0104
-6031 877
+ •108116
-•0136
4-0233
-6045 824
+ -107869
- -0130
4-0294
-6052 404
The 'divided differences are as shewn. Thus approximately
log 4-01 = -6020817 + -0098 x -108431 + -0098 x -0004 x -0136
= -6031444,
which is correct to seven places.
The error due to the remainder term is of order
-0098 X -0004 X -0133 x -4343 x 2
cc3x3!
DIVIDED DIFFERENCES
7
1-2]
where x varies between 4-0002 and 4*0294, which is less than 2 in the
10th decimal place. The above value could therefore be affected
only by errors of rounding in the seventh place.
1*3. The Divided Differences are Symmetric Functions
of the Arguments. By definition
[1=1,]=®-+/^),
1- XJ Of* ry* <7*
Jj U/j Jj
SO that we obtain without difficulty
\xxx^= I I .
It is now very easily proved by induction that
(1) [xx^x^.-.x^]
/W ^ .
' {x-oi>j){x-x^) ... {x-x„) {Xj^-x)(xj^-X2) ... (a%-a;„)
^ fi^n)
{x„-x) {x„ -Xj)...{x„- a!„_i) •
Clearly the interchange of any two of the arguments does not alter
the value of the divided difference, which is therefore a symmetric
function of its n arguments.
For example [^^1^2! ~
Again [x^x^x^ . . . x„_jX„x„+i] = [x^x^x^ . . . x„.^jXjX„+j] ,
SO that
{x^x^x^...x.^'\-ix^...
^n^w+l]
V^) -
*^1
— [^n^2^3 *•* ~ [^2^3 ***
^n+1
~ [^1^2^3 * • * ^w-l^n+l]
1*31. The divided differences of x" can be obtained as
follows :
from 1*3 (1),
p^i
[Xj^ . . . x„+j] - (x, - a^) (a;^ -x^) ... (x, - '
DIVIDED DIFFERENCES
[1-31
This last is the coefficient of in the expansion of
t {x, - a^i) . . . {x, - (1 - xj) {x, - x,^^) ...{x,-- x^^^) *
But this expression is evidently the result of putting into partial
fractions the function
(1 -- - x<^t)-'^ ... (1 -
and hence the coefficient of is the sum of the homogeneous
products of degree n-f of ^25 •** ’ ^p+V
Thus [x^x^ . . . J = S Xi' ^
where the summation is extended to all positive integers including
zero which satisfy the relation % + a2+ ... ~ n-f.
For the divided differences of - we have
X
1
Hi i^s - a^i) . . • {^S - ^s-l) - ^s+l) • • •
and this is the value when ^ = 0 of
- y t
Hi i^s - ^^l) - ^s) {^S “ ^s+l) • •
which is obtained by putting into partial fractions
-1
i)’
so that
r 1 (-1)^
X^X^ , . . Xpj^-^
1*4. Lagrange's Interpolation Formula.
From 1-3 (1)
we have
(1)
where
f(T\ =z fir \ 3:2) (x x^...{x — x^
.rz-vX (X!-Xj){x-Xs)...{x-X„) ,
R„ {x) = {x-Xj){x-x^) ... {x- x„) [xoojX^ . . . a:„].
''G. Chiystal, Algebra, 2nd edition, (London, 1919), 205.
1-4]
DIVIDED DIFFERENCES
This is Lagrange’s Interpolation Formula with the remainder
term Comparing with 1*1 (2) we see that this remainder
term is the same as the remainder term in JSTewton’s Formula. It
follows that Lagrange’s Formula has exactly the same range of
application as Newton’s and yields identical results. .
The formula may also be written in a slightly different form.
Write
(2)
(3) Then
^{x) = {x^x^){x-x>^) ... {x-x^).
/w
t^xX-x, (f)'(x,)
1 *5. Expression of Divided Differences by means of
Determi nants. By a well-known theorem in determinants origin-
ally due to Vandermonde and generalised by Cauchy, we have
(1)
1 1 1
^3
^1^ ^2^
1
n— 1
= n - «=■)>
j>i
where the product expression has \n{n~\) factors. This import-
ant determinant is usually called an alternant.
Now from 1*3 (1),
[XjX^s - = 2
fi^,)
=1 - ^i) (a;^ - {x, - x,+^) ...{x^-x^)
S=1 j i j>i
where YV - x^ means that the value s is not to be ascribed to the
suffixes % j. Now
2(-l)"-'/(x.)n'(a:,-x,)
/(^)
/(^2) •
• fi^n)
1
1
1
X2 .
/y. n-2
'^2
/y ra— S
as is evident when the determinant is expanded by its top row.
10 DIVIDED DIFPEKENCES [1-5
Hence rearranging the order of the rows and thereby removing
the factor ( - we get
(2)
li^l)
/(*«)
. n~l
, n— 1
n—l
n
^ n-2
^^n-2
^ n—2
•^2
n—2
n
rp n-3
^^n-3
/yt fl — 3
Xg,
rf fl— 3
%
X2
1
1
1
1
1
1
1-6. Divided Differences expressed by Definite Inte-
grals. We shall prove by induction the following formula, which
is due to Hermite :
1*1 fii f'„_2
(1) [xjX^ ... a:„] = dii dt^ ...
Jq Jq Jq
where U„ = (1 — ifj) — ^2) ^2 (^n-2 ~~ ^n—i) ^n-l ^n-i^hf
and ^1, ••• ^ ^n-1 3.re to be treated as (n- 1) independent variables,
which of course disappear when the repeated definite integral is
evaluated.
Froqf. When w = 2, the right-hand member of (1) becomes
f / '{(1 - ih) % + .
Jo *^2 ”
so that the formula is true when n~ 2. We assume it to be true
for n arguments, and proceed to employ a new x^j^i and a further
parameter
Now
j* 1) ((f ~ ^x) ~b ♦ — 4- (^n-2 ~~ ^w-i) ^n-X + ^n-l^n)
0 ^n~~~ ^n+1
_ ^1+ - ‘ + (^w-2“" ^w~l) ^n-i +
Hence f dt^ f
•Jo -J n
_ [XjX^ ... x„] - [x^x^...
1-6]
DIVIDED DIFFERENCES
11
_ \PA ... - [^^2^3 ... ^n^n+l] ^2)
^1 ^71+1
= ••• >
SO that the result follows by induction from the case n = 2.
1*7. Divided Differences expressed by Contour Inte-
grals. Consider a simple closed contour 0 enclosing a simply
connected region of the complex variable t in which are situated
the points Then by Cauchy’s Eesidue Theorem,* if
/(i) be holomorphic * throughout this region and on the contour C,
Again, the residue — of the function
, /W is
— — ... {t — Zn) (2s~%.) ••• i^s~^s-l){^s~^s+l)
Hence t
(1) Jl- f -/(Q- V'
[^1^2 **• ^n]} ^y (1)5
which is the required expression by a contour integral of the divided
difference of order n-1 of f(z}.
We can use this result to obtain another proof of Newton’s general
interpolation formula, but with the remainder term now expressed
as a contour integral. We have
1 1 Z-Zj^ 1
t-z t-z-^ t-z^t-z^
^ I
t-z t-z^ t-z^t-z'
* See 84 and Whittaker and Watson, Modern Analysis, 4th edition,
(Cambridge, 1927), 5-2, 6*1, 5-12. This work will be cited in later footnotes
as Modern Analysis.
tThe notation S' means that the factor is excluded from the
denominator for ^ = 1, 2, , n.
12
DIVIDED DIFFERENCES
[1-7
SO that by repeated substitution for
— ^ we set the identity
t-z ^
L = i- + — — I- + - J— + . . .
4. J
(i-z{j[t-z^) ..."(t-z^y t-z’
so that .
;^-2: 2’:ziJot~-z^ ^ ^ 27zi J c (t - (t ~ Zq)
fity
{t-Zn){t-z]
that is
(2) f{z) = f{Zi) + (3 - %) [%Z2] + (z - Si) (2-22) [212223] + • • •
+ (2 - Sj) (2 - 22) ... (s - 2„_i) [2i22 . . . Z„] + i?„ (z),
where
(3) R.M = (»-'.) 2^ j„ (r-i,)'®‘V-^j f-.'
which is again IsTewton’s general formula. But it should be
observed that, while in 1-1 (2) f{x) is unrestricted, in the present
case f{t) is an analytic function holomorphic in a certain simply
connected region.
1*8. Divided Differences with Repeated Arguments:
the Confluent Case. The identities yrhich define the divided
differences in 1-0 become indeterminate if two of the arguments
coincide. By 1-2 (2) we have
(1)
(W-I)l
where 7] Kes in the interval bounded by the greatest and least of
ccj, a’2, ... , Xn. If all these variables coincide with we take, as the
definition of ••• 2*1], the value of the right-hand member, so
that for 71 coincident arguments x-^
[x^x,...x^]
(71-1)1 •
(2)
1-8]
DIVIDED DIFFERENCES
13
The limiting value of a divided difference, which arises when
two or more of the arguments coincide, may, with propriety, be
called a confluent divided difference arising from the confluence of
the arguments in question.*
Provided that we write the difference scheme in such a way that
all the arguments coincident with a given value occur in a single
group, we can form a complete scheme of divided differences by the
use of (2) above and the definitions of 1*0 without encountering
indeterminate forms. Thus
Xi
M)
U"(x^)
f (*i)
U'A^i)
x^
/(*i)
Z'K)
[X^X^X^X2]
fi^i)
[x^x-ip:^
[^V^z]
A^z)
[xjxa^
f'i^z)
[XjX.2X.^X^]
A^z)
W’i^z)
f'i^z)
A^z)
[a'^a's]
A^z)
In this scheme, for example,
which is perfectly determinate.
In the case where all the arguments coincide .with
Newton’s formula 1*1 (1) yields Taylor’s expansion, namely,
J{x) + /"(^i)+-
"■‘'cf. Modern Analysis. 10*5.
14
DIVIDED DIFFERENCES
[1-8
R^{X) =
nl
where ? lies in the interval (x, Xj).
It should be noted that confluent divided differences can only
be formed if /(a?) possess the necessary derivates.
To obtain a formula for when arguments are equal
to Xj, % arguments are equal to arguments are equal to
Xj,, we use 1*7 (1), which gives the interpretation
= 3^ I =
1 1 1 r f{t)dt
(«1- 1)! (Wg- 1)! "■(«„- 1)! Jc
__ 1 1 1 0»i+W2 + ...-fnp-37 ^
^ ’ ~ K-l)! (Wg-l)! ■■■ K-1)! dxj^i-K..dx^”P-^
If all the n arguments coincide with x^, we have
in agreement with (2) above.
m
it =
(w-l)!
1-9. Interpolation Polynomials. A polynomial of degree
ra - 1 at most, whose values at the points a^, ajg, , a;„ are the same
as the values of given function/(a;) at these points is called an inter-
polation polynomial of /(a:). If I^^{x) denote such a polynomial,
we have at once from Lagrange’s interpolation formula 1-4 (3),
(1)
where
In-Ax) = y] -iM. _ y
M x-x, cl>'{x,) ^^x-x,j>'{x,)
4>{x) = (x-Xj)...(x-Xn).
It is clear from this result that the degree of (x) is at most n - 1 .
Only one such poljnomial with given agreement can exist, for if
Jn-iix) denote a second polynomial with the same agreement, the
polynomial I„-Ax)-J„_Ax}, which is of degree w-1 at most, has
n zeros x^, x^,...,Xn and therefore must vanish identically.
To recover the interpolation formula from which (1) wa^ derived
we have sunply to add the remainder term
Rn {x) — [X X^ (X-X^) ...(X- X„) [XX^X^ X„].
DIVIDED DIFFERENCES
15
1-9]
Thus we have
/(») +
Since Ifewton’s interpolation formula has the same remainder
term as the formula of Lagrange, we have, from 1*1 (1), the alter-
native expression
n-l
(2) 1 „_i (a:) = f{x^) + 2 (a: - Xj) {x-x^)...{x- x,) [x^x^ . . . x,+j].
S=1
For example, if n = 3,
h (^) = A^i) + (^-^) [xjx^] + (x-xi)(x- x^ [XjX^x^l
It should be observed that an interpolation polynomial, being
fized by the values of the function at the given points, does not
depend on the order in which these- points are considered. Thus if
we take the four points x_j^, Xg, x^, x^ in turn in the orders
^o> ®-a> ^2 a.nd x^, Xg, x^, x_^
we have the two expressions
hip) =J{^q) + (35 - aJo) [a:oail+ {x - Xg) {x - x^ [a:oa;ia:-i]
+ {x- Xg) (x -X^){x- a:_i) [x^x_.^X2l
h (^) = /K) + (» - IPi^olt + (a: - »i) {x - Xg) [x^XgX^]
+ (x-x^){x- Xg) (x - Xz) J.
Adding these expressions and dividing by 2, we have
(3) I a {x) = i {/ (a^o) +/(ai) } + (a? - Jxj - ^Xj) \Xfp{\
+ {x — Xg) (x — Xj) { ix^jXgfXjJ + [Xga^Xg] }
•+ (x - Xg) (x - Xi) (x - ^x_i - Jxa) [x_iXoXiX2] ,
which employs the divided differences shown in the scheme
x_i
^0
[XoXj]
Xz
fi^o)
fi^)
[x_iXoXj
[XoXiXa]
[x_iXoXiX2]
16 DIVIDED DIFFERENCES [1-0
From I^ix) we could obtain an interpolation formula by adding
the remainder term
Ri {x) = (x- a:_i) (a: - Xg) (x -x{){x- x^) [xx_jX^jX^].
Again by taking five points x_^, x_^, Xq, Xj^, x^ in each of the
orders Xq, x^^, x^, x^^, and x^, x^, x_i, Xj, x_2 we obtain two
expressions for I^ix) whose arithmetic mean gives
(4) h(x:)= fi^o) + (a: - a^o) H [a^-ia^o] + [a^o%] }
+ {x — sIq) {x —
+ (® - a;_i) {x - Xo) {x-xj}i{ [a:_2a:_ia:oa=i] + [a5-ia:oa:ia;2] }
+ {x- X_i) {x - Xo) {x - Xi) (x - |-X_2 - Jxj) [x_2X_iXoXiX2] ,
which employs the divided differences in the scheme
a;_2
a:-i
•
1 — 1
0
V
a^o /(a=o)
[XflXi]
1 1
0
Xj
X2
From I^^ix) we
could obtain an interpolation formula by adding
X'JLUULL A 4.
Bs(x).
The above results, (3) and (4), are also due to Newton. They can
easily be extended to include divided differences of any order, the
form (3) being taken if n be even and the form (4) if n be odd.
Eeturning to (2), if two or more of the arguments coincide we
obtain a confluent interpolation polynomial. Thus if ^ = 4, with
the arguments x-^, ^2, x^, we obtain
h{^)=f (%) + {x-Xj) [XiXj + (x - Xj)^ [XjP^X^]
+ (x — Xj)^ (x — X2) [Xj^X2X2X3]
=/(ah) + (a: - a^) /' (x^) + (x - x^)®
+ (x — Xj^)2 (x - Xg) [XjXjXg] ,
i.9] DIVIDED DIFFEBENCES 17
SO thatj in this case,
Iq{Xj) 1 2, =/
It is easily seen, in the same way, that if v arguments coincide
with then
(5) f"’’ {x^) = (Ki), s = 0, 1 , . . . , V - 1,
and the polynomial may be said to have agreement of order v with
the function f[x) at the point x-^. In this way we can construct
polynomials having arbitrarily assigned orders of agreement with
the function at given points. Thus the confluent interpolation
polynomial of degree 4, which has agreement of order 3 at and
of order 2 at is
^4 (®) = fi^i) + (2: - aji) [xjxj + (x- Xj}^ [aJiaJA]
-h(x- X^f [^iX2XiX2] + (x- (x - X2)
= /(%) + i^- ^1) fi^i) + 2^
1 92 19®
+ i^- ^1)^ m 9“2 [^1^2] +{^- ^1? - 3^2) 2 1 ^2-9^
That this polynomial has agreement of order 3 at is obvious.
That the agreement is of order 2 at Xg is equally obvious if we observe
that the polynomial could have been written down in an alternative
form with the arguments taken in the order X2X2XjX^Xi .
EXAMPLES I
1. Shew that the divided differences of f{,x)-{-<f){x) are the sums
of the corresponding divided differences off(x) and of (j>{x).
2. Shew that the divided differences of cf{x) where c is a
constant are c times the corresponding divided differences otf{x),
3. If the arguments be each multiplied by the same constant c,
while the tabular values remain unchanged, shew that the divided
difference [x^X2 ... x^^^J is multiplied by c~”.
4. Shew that the divided differences of f{x) are unaltered if the
arguments be each increased by the same constant c, while the
corresponding tabular values are left unchanged.
jg DIVIDED DIFFEBENCES [ex, ]
5. Form the divided differences of the polynomial
+ ... +a„.
6. Prove that
. . . 2^«] = I . . • + ^2^2 + • • • + ^n^n) • dt^ ,
where the integration is extended to all positive values, including
zero, which satisfy i5i + ^2+ ... = 1. [Genocchi.]
7. With the notation of 1*8 (3) shew that
where
[x^x^.,.x^x^~\- f dk r dk... [
Jq Jq Jq
2/ — (1 — ij) a^+ ig) ^2 + . . . + (fj,_2 ip-i) >
, .. (1 - «ir~' ih - hr~' - fe-2 - cv '
K-l)!(n2-l)!...K-l)!
8. If
L M = (g - a;i) (a - jgg) . . ■ (a; - a;,_i) (x - x,+^) ...(x-x„)
{^r~ ^l) {^r - ajg) - {^r “ " ^^r+l) • • • i^r “ ^n) ’
r = 1, 2, ... , n,
prove that
L^(x)-\-L2{x) + ...+Ln{x) = 1,
ixi-xYL^{x) + (x^-xyLs{x) + ... + (x„-xyL„(x) = 0,
V = 1, 2, 3, ... , n-1.
9. Prove that the function
t-g {t-a){t-b)
a-b {a-b){a-c)
becomes unity when t = a, and zero when t = b, c, .
Hence with the notation of example 8, prove that
x^-x^ ix^-x^){x^-xs) (a:i-x„)’
witli similar expressions for L^ix), Lg{x), ....
10. Deduce Lagrange’s form of the interpolation polynomial
from tke rule for resolving
fM
(x-a^)(a;-a;2)...(x-x„)
into partial fractions.
DIVIDED DIFFERENCES
19
EX. 1]
11. Find the polynomial of the lowest possible degree which
assumes the values 3, 12, 15, - 21 when x has the values 3, 2, 1, - 1
respectively.
12. Three observations u^, of a quantity are taken near
a maximum or minimum. Shew that the value of x at the maximum
or minimum is approximately
(62 - c2) Ua + (c^ - a^) Ui, + (a^ - b^)
2{{b-~c)Ua-\-{c-a)Uf,-{-(a-b)Uc} ‘
13. The values of a function at m + n points are given. Prove
that a rational function, whose numerator is of degree m-l and
whose denominator is of degree n, may be found, which assumes the
m + n given values at the given points. [Cauchy.]
14. If m = 2, ^ = 1, prove that the rational function of example
13, which assumes the values u^, at the points a, 6, c, is
~ UjUc (6 -c){x-a)- u^Ug (c -a){x-b}- UgU^ {a-b){x-~ c)
Ua (6 -c){x-a) + (c - a) (a: - 6) + {a -h){x-c)
15. If the function
(a;) = ^ 0 + cos x + sin cc) + . . . + (^ „ cos nx + sin nx)
assume the values Wg, , W2n+i when x = x^, Xg, ... , a?2n+i5 prove
that
^ sin i(x- x^) sin ^(x-x^)... sin jjx- ^
^ S = ] sin J (x, - a;i) sin ^ (x, - Xg) ... sin (a;, - ® ’
the factor which becomes indeterminate when x = Xg being omitted
in each term of the sum. [Gauss.]
16. By means of 1*5 (2) express the confluent divided difference
[aabc\ in the form
m f{a) m f{c) 3a2 6^
2a 62 ^2 a2 2a
d 1 b c ’ a \ b 0
10 11 10 11
17. Express the confluent divided difference [aaa66c] as the
quotient of two six row determinants.
18. From the confluence of the n arguments in 1*5 (2) deduce
the formula 1*8 (2).
CHAPTEK II
DIFFERENCE OPERATORS
2‘0. Difference Notation. Let J a; be an increment of the
variable x. The corresponding increment of a function* u{x) is
then given by
J u{x) = u{x+^ x)-u{x).
This increment ^u{x) is called the jSrsi difference of u{x) with
respect to the increment /\x. The most important case arises
when the increment J a; is constant. Denoting this constant by w
we have for the first difference of m(x)
(1) m(x) = w(x+o))-u(x).
The result of performing the operation denoted by the operator J
is still a function of x on which the operation may be repeated. We
thus obtain the second difference
/^u{x) = /i[/iu{x)] = ['i4(x+2cd)-w(x+co)]-[w(x+c!)) — w(x)]
(2) J^'ii{x) = «(x-f2a))-2M(x+a))+w(x).
Proceeding in this way we can form the third, fourth, nth
differences, namely,
A^u{x), #m(x),...,
by means of the relation
A'uix)=A[A^-'^u{x)].
We find, for example, that
A ^ = +
zj^a;3 = 6x0)2 +60)3,
z13x3 = 6o)3,
A^x^ = 0.
*\]e shall denote a function of a: by u(z) or by according to convenience.
20
2-0] DIFFEKENCE OPERATORS 21
The successive differences of a tabulated function are easily
formed by simple subtraction. Thus for the function we have
X
A
A^
A^ A*
0
0
1
1
1
7
6
6
2
8
12
0
19
6
3
27
18
0
37
6
4
64
61
24
125
More generally, if we denote the functional value + by
we have the scheme
Argument
Function
3
I
W_2
A “-2
a - 0)
U_i
zJw-i
a
Uq
A^u-i
Zl*^-2
A ^0
(X -f* G>
tq
Zl" Wp
zl%
Zl^Wo
^2
A ^2
A^^i
G H“ 3g)
where each entry in a
vertical difference colunin is obtained by
subtracting the upper entry immediately to the left from the lower
entry immediately to the left.
By adjoining further functional values we can extend the scheme
as far as desired. Inspection of the scheme shews that to form a
fifth difference six consecutive tabular entries are required. Simi-
larly, to form a difference of the ^'ith order, n + 1 consecutive
DIFPEBENCB OPERATOBS
[2-0
tabular entries are necessary. In the above scheme the differences
■which lie on a line sloping diagonally
downwards from Mq, are called descending, or forward, differences
of Ug. The differences /j u_^, ••• , which he on a line sloping
diagonally upwards from Ug are called ascending, or backward,
differences of Ug.
2*01. Central Difference Notation. If we introduce the
operator S defined by
the difference scheme of the last section becomes
3
1
U^9
a-co
Su_^
Su^
CJ + O)
%
Su.2
^2
Sus
a+3co
Bhig
S%o
ss
The operator * S is the central difference operator and the differ-
ences in the above table are known as central differences. It should
be carefully observed that the numbers in the above difference
scheme are the same as the numbers in the corresponding positions
in the scheme of 2-0. The two schemes differ only in the notations.
It will be seen that Bu^ = v^-Ug, = and so on.
The differences in the same horizontal line with are labelled
■with the s'ufiix h. Those on the horizontal line between Mj, and
Wi+i are labelled with the sufiSx A+J. The notation of central
differences is useful for the compact description of certain inter-
polation and other formulae. The arithmetic mean of successive
* This notation is due to 'VV. F. Sheppard, Proc. Land. Math. Soc. 31 (1899),
DIFFERENCE OPERATORS
23
2-01]
differences in the same vertical column is denoted by and is
labelled with the arithmetic mean of the suffixes of the entries
from which this expression arises.
Thus
|(8X+SX) ==
When these are entered in the difference table the lines a, a + o,
and the line between, will have the following appearance.
Mfl
S%o
sx
Swi
where denotes +
Another notation, originally due to Gauss, for central differences is
(m, n) where m denotes the row, and n the order, thus
§2^0 = (O3 % = (i 5)-
2’1. Difference Quotients. The notations of differences ex-
plained in the preceding sections, while of the greatest practical
utility, do not sufficiently unmask the close analogy between the
finite and the infinitesimal calculus. We now introduce Norlund’s
operator A? which is defined by the relation
(1)
We call which is evidently a divided difference, t]xe first
it)
difference quotient of u{x). This symbol has the advantage that
(2) lim A^(^) = Du{x),
(O — >-0 0)
where D denotes the operator of differentiation, in this case djdx.
The operation can be repeated, thus
, ^u{x + o)}- /^u{x)
Ku{x) = A[Au{x)] = ^
(t) it) (t) CJ
u{x-{-2(ii)-%u{x + ui) + u{x}
(3)
24
DIFFEBENCE OPERATOBS
[2*1
and generally for the ?^tli difference quotient
n
A «(;»)= A A •
a> w L_ 0) —
From this we infer the useful relation
n n + l n
AM(a; + Co) = 6) A w(a!)+ A
CO <ti a>
We have also
n
(4) lim Aw(a:) = D"M(a:).
CO“->0 CO
From the definitions it is clear that the operators /J and are
related by the formula
(5)
CO
and in the special case where ca = 1 the two operators have precisely
the same meaning.
If 6) = 1, we shall write A instead of A*
1
2*105. Partial Difference Quotients. Consider/(a;, w) where
X and u are regarded as independent variables. Let x be given the
increment o>, and u the increment A. We then define ‘partial differ-
ence quotients with respect to x and u by
Axj{^3 w) = [/(aj+co, u)-f[x, u)]j CO,
An fix, u) = [f(x, u + h) -fix, U)] I h.
The difference of f{x, u) is defined by
Af{:x, u) =/(a;+co, u-\-h)-f{x, u)
—fix+o}, ^ + u + h)+f{x, u + h) -fix, u)
=f{x+o), u+Ji)^fix-\ro>, w)+/(a;+co, u) -f{x, u).
Thus we have the two equivalent relations
0-) Af{x,u)=:c^ Ax fix, + h Aufix, u),
w h
(2) AM ^) = CO Affix, U) + hAuf{x + 0,, u).
2-105]
DIFFERENCE OPERATORS
25
Again
Aa/(a^> “) = [A«/(a:+«> w)- A«/(a:, «)]/«
= [/(a:+<d, m + A) -/(a;+6),M) -f{x,u+'h)-{-f{x, u)] /
The symmetry of this result ia h and to shews that
(3) Ax Auf{x, u) = A« Ax/(aJ> «)•
u) h h ui
2*11. The Difference Quotients of Factorial Expres-
sions. Products of the forms
(1) u(x) . u{x-Ci)) . ^^(a;~2ca) ... w(^^-mco4-co),
(2) w(a? + 6)) . 'i^(a;+2a>) . ^(a; + 3co) ... u{xi-m(x))y
where mis a positive integer, are called expressions, the first
being a descending factorial, the second an ascending factorial ex-
pression.
Of expressions of these t3rpes the two simplest and also the two
most important are
(3) = rr(ir-co)(a;-2co) ... (ic-mo-f co),
(4) = (a;-f-(o)-i(cr+2o))-i(a; + 3co)-i ... +
If m = 0 we interpret each of these expressions as unity, that is
and if 6) 0, we have also
lim = x^,
(u — )- 0
lim ~ x~'^.
To form the diiSference quotients of we have
_ (a; + co-a;+mo>-co) x{x~<x) ... (a: - ?nco + 2co),
Hence, if n^m,
n
_ m(m-l) ... (m-w+1)
26 DIFFERENCE OPERATORS [2.11
which can also be written, after dividing by m !, in the form
(5)
A-
7?i! (m-n)!'
If 71 = m, we Iiave
n:<m.
m ^(ma>)
while if n>m the result is zero.
If 0) = 1, we have
7n! m!
=(*)
\m/
in the usual notation for Binomial coeflicients, so that (5) yields the
important formula
(6)
A ^
^ Vm/ \m ~ nJ
n^m.
Again from (5), if m ^0, we have by 2*1 (4)
m! (7n-7i)\‘
These results shew the analogy between in the finite calculus
and in the infinitesimal calculus.
For the difference quotients of we have
a;+6)-a;-ma)-co
a, (^+o))(a;-f 2co) ... (a;+m<o + co) ’
j^X^~ = — 77lX^ ~m(o- to))^
ti)
so that
n
(^) Ak (“‘^■“1) ... (“-971 — ~
which can be written
n
w ^
When 6) O5 we have
D^{m-l)lx~‘^ = (- + 1)!
DIFFERENCE OPERATORS
27
2-11]
For more general forms of the types (1) and (2), we easily obtain
CO /\ • • • 'li/g -moi+bi
Oi
“ ^X+bi ~~ "^X - wto+w) '^X '^X - <0 • • • '^X ~ J»a)+2a>
1 ^x-i-b) '^x-hmcit+co
oi ^x-{-b>^x-\-Zb3 •*• ^ic+7Ww ^x~h(o^x-j-2(t} •*•
In particular for Ua, = ax + by
we can write
(8) Ux'^x—bi *•• '^x-'m(o+o> ~ (ci2/ + 6)
(9) : (aa;+ 6)
^a:+oii'^a;H-2to • •• '^x+moi
and we have
(10) A
(11) +
2*12. Expansion of a Polynomial in Factorials. Let
<l> (x) be a given polynomial of degree m. Assume that
2j(to) aj(2w) 2j(rrta>)
(1) ^(x)^ao+«i-Yr+®2'2r + - + ®™ 1^’
which is evidently a legitimate assumption since the right-hand
member is a pol3aiomial of degree m with m+ 1 arbitrary coefficients.
Forming the successive difference quotients, we have by 2*11 (5),
aj(Wa»~aj)
Afi^) = % + (m~ ij! '
= 02+^3 -Yr + - + °”» (m-2)!'’
m
l^4>{x) =a^.
(j3
If in these results we put x — 0, we have expressions for the
coefficients in the form
A<i>{% 5 = 0, 1, 2, ... , m.
bi
28
DIFFERENCE! OPERATORS
[2-12
Thus
/y.(w) 3j(2to) 2
(2) ^(a:) = ^i(0) + -Ty A'^(0)+-9rA^i(0)+-” +
i- • tu ^ ‘ (O
Q^iniui) m
ml
, A^6(0).
The coefficients in this expansion can be obtained by writing down
the values of <f>(x] for a: =: 0, w, 2w, , mco and then forming the
successive difference quotients. Thus for
<j)(x) = + + to®,
we have
X
<f>{x)
/^<j>{x)
A4>{^)
0
£0®
46)3
0)
5co*
10o)2
6co
2a>
15u)S
226)3
12ca
3co
376)3
so that
3
A<f>(^)
6
3cii^ x+co^ = a>^ + 4co^x^"^ + 3ca +
Another method follows from observing that the coefficients
ttQ, ... , are the successive remainders when we divide cl>{x)
by X, the quotient of this division by (x-co), the new quotient by
(x-2a>), and so on.
Thus with ^(x) = x^ + 3a>^x4-co^ we have
X j X^-r3co^X + CO’^
X - oj X- + 3co^ remainder co^
X - 2co X + o) remainder 4a)^
remainder So)
which gives the same expression for (f>{x) as that obtained by the
first method.
2-13. The Successive Difference Quotients of a Poly-
nomial. To obtain the successive difference quotients we can
express the polynomial in factorials by the method of the preceding
2-13]
DIFFERENCE OPERATORS
29
paragraph and then apply 2' 11 (5). Since each application of the
operator A a polynomial lowers the degree by unity we have the
following important theorem :
The mth difference quotients, and also the mth differences, of a 'poly-
nomial of degree m are constant. The differences of order higher than
the mth are zero.
Thus with the polynomial
we have
/S.(f>{x) = 60)33^*“) + 4oa^ = -h 3cox + 4co^5
2
(ti
= 6,
CO
A<i>{x) = 0.
G — (1 “i"
we have
(2) A(l + M" = ^”(l + &co)"
Since
X
lim {l + 6o>)" =
co-»0
we have as a limiting case of (2)
X
Thus in the finite calculus (l+co)^^ plays the part of e*.
DIFFEKENCE OPEKATORS
30
2*2. Properties of the Operator A-
[2-2
From the definition
it is evident that ^ obeys the following three laws :
(i) The distributive law
(ii) The index law
mrn ~| w+w n r m “■
A - A «(:») = A Aw(a:) ,
— * <i) 10^0) —
where m and n are positive integers.
(iii) The commutative law with regard to constants
l^cu{x) = c l^u{x),
(t) 0)
where c is independent of z. This result is also true if c be replaced
where &{x) is a periodic function of x with period co ; for
(oA®(a:)w(a:) = m[x+oi)u{x+a)- ■b3{x)u{x)
(1)
= m{x)u(x-ho^)-rn{x)u{x)
= o^VLf(x) J\u{x).
to
If then ^(X) = + be a polynomial in X whose
coeflhcients are independent of z, we can associate with an
operator ^i(A)j such that
to
= aoAu{x) + a^”^ u{x) + ...+a„u{x).
^ to (jj
If (l>2{k) = + be a second polynomial, and if
we expand their product in the form
(h) <l>2 W = ^ ^
we have, on account of the above laws,
<k{A)<f>2{A)u{x) = 42{A)MA)u(z)
m+n «if n - 1
= CoA^(aj) + Ci A +
DIFFERENCE OPERATORS
31
2-2]
We may also note that the above results are still true if the
coefficients of the poljmomials be replaced by periodic functions
of X with period co.
2*3. The Operator V- The definition of this operator is given
CO
by
S] u{x) = -J [u{x)^u{x-\-(^)],
CO
which may be compared with the definition of the central difference
averaging operator [x of 2-01.
Repeating the operation, we have
2
y u{x) = I [u{x) + 2u{x+(^)-{-u{x-\-2c^)] ,
CO
and generally, as is easily proved by induction,
Y»w = r,
+...
As an example,
u{x) + (^^'ju{x+(j:)) + {2) +
+ (^)w(a:+«w)]-
ya® = |a®(a"4-l),
Va* = ^a^(a“+1)"-
2*4. The Operator E". This operator is defined by the
relation
(1) = u{x + o>).
The operation may be repeated any number of times. Thus
The operator E“ clearly obeys the same laws of combination as A-
60
In particular, if <f>2Q^) be the poljmomials of section 2*2
above, we have
^i(El9^2(E“)^(^) = <I>2{E-)ME^Mx)
= CoE U (x) + u{x}+... + c„+„u(x)
= CQu{xi- m(x> fuo) + CjU{x+mo) + no> - o^) + ... +c„j+„w(a;).
32
DIFFERENCE OPERATORS
[2’41
2-41. Herschel’s Theorem. If f(X) be a 'polynomial 'with
constant coefficients and if ~ then
He-*) = Hi)+tHE“)o+^^,HE-)o^+... ,
or symbolically
cl>{e-^)^<f>{E^)e^-K
The sign = is used to denote symbolic equivalence.
We have (^(e“*) = and it is therefore sufficient to prove
the theorem for <f>{e-t) = e«“<, for the result will then follow by
addition of constant multiples of terms of this type. Now
gix+nu,)t _ l+i;(a:+wco) + ~(a: + Moi)2+...
= l + iE"“a: + 5^ E"“a:N-... .
Putting cc = 0, we have
f2
gn^t _ l + iE«"0 + |^ E""0’*+...
which proves the theorem.
2-42. From the definition of E“, we have
E"“ a® = = a* . a”".
Thus if <f,{X) = be a polynomial in X, we have
^(E“)a®= = a^f{a'“).
More generally, if the power series
9^(X) =
w=0
be convergent for X = a", we have
9^(E“) a® = o®^(a").
2-43. Theorem If f(X) ig a polynomial ■whose coefficients are
independent of x, then
<i>{E'‘)a==u{x) = a=‘<j>{a''‘E“)u{x).
9^(E")w(a;) = a“,J(a“E“)a-®w(a;).
Let 9S(X) = SM„X”.
2-43]
DIFFERENCE OPERATORS
33
Then
u{x) = 0'^u{x)
~ w(a3 + noo)
= a^^(a‘“E‘^)^(3?),
which proves (1), and (2) follows by replacing u{x) by a'~^u{x).
2 ‘5. The Relations between E" D. We have from
the definitions
E“«(a:) = M(a;+co) = M(a:) + 6)
Thus
E“ = l+e)A,
coA=E”-r.
As deductions from these relations, we have Gregory's Theorem,
namely,
(1) u{x^n(j^) = = (1 + 0)
ui
= u{x) + (^o^ ^u{x) + (^(^^ l^u{x) + ...
+ l^u{x) ,
n being a positive integer. This formula expresses u{xi-no^) in
terms of u (x) and its successive differences.
Again, we have
(2) o^^^u{x) = (E'^-^^uix)
(U
= u{x+no^)-'(^u{x-j-n-lo^) + (^u{x + n-2c^)-
which expresses the nth difference in terms of functional values.
Again, assuming that u{x-\~<h) can be expanded by Taylor’s
Theorem, we have
z= 'u(cc+6)) = ^^(^r) + coD w(a5) + ^D2t^(ii?)+ ...
ize^^u{x).
34 DIFFERENCE OPERATORS
Thus we have the relations of operational equivalence
l+0!)A = e"^,
so that
n
A “ («) = (e"-® - 1)« M (x).
2»51. The Analogue of Leibniz’ Theorem. The theorem
of Leibniz in the differential calculus, namely,
D^{m) = (D«m) v+(^{B^--i-u)Bv+Q{D^-Zu)I^v+ ... ,
where D denotes the operation of differentiation, has an analogue
in the finite calculus, which we proceed to obtain. We have
(1)
= {(EEi)“-1}m«u*,
where the operator E acts upon m* alone and the operator E, acts
upon alone. Thus we have
(1) «"A = {{E Ei)“ - 1}» .
In the ezpressions of E and Ei let us suppose that A acts on
u* alone, while Ai acts on u* alone, so that
E" = 1 + coA, Ei“ = 1 + 0)Ai.
Then we have
(EEi) ^ ).
Thus " « u.
A(M,«.) = fA+E“Aa)»M
" « «
2-51] DIFFERENCE OPERATORS 35
Bemembering that A and E operate only on we may drop the
to
suffix and write
(2) A {Ux%) = (a %) + ( i) ( A «*+„) A %
which is the required theorem. Since
s
lim
w <*»
we see that Leibniz’ theorem may be regarded as a limiting case of
this result.
The theorem may be expressed in other forms. If in (1) we expand
the right-hand member directly, we have
(^) ^aj+nar^a;+n(«> ^ ^aj+{n— l)w^aj+(n--l)a»
which is in fact a case of 2*5 (2).
If the expansion be required in difference quotients of Ug. and
we write
(EEi)”*-! =w A+wAi+“® AAi,
<lt Ui
so that
= (A+Ai+“ AAi)”K^’x)-
(il w a> ut w
The expansion of the right-hand member gives the required result,
but it is hardly worth while to write down the general expansion.
2‘52. The Difference Quotients of a’' v^. By 2*14: we have
A = ^aj+sw
h =
where
36 DIFFERENCE OPERATORS [2.5
If then in (2) of 2-51 we put = a* we obtain
= «*[— ^+“"Aj ‘»z‘
Thus if <j>[l) be a polynomial, we have the operational theorem
94(A) a^i'x = a* 94 (a"A+ u, .
<0
If next we put a“ = 1 + aco, we obtain
K X
^ [A] (1 + «<o)“ V* = (1 + aw)“ 94 [( 1 + aco) A + a]
If we now let w 0, we obtain the corresponding theorem for the
operator!), namely
94 (D) = e«’^94(D + a) v
2-53. The Difference Quotients of Zero. The value of A
when a: = 0 is written A O'" and is called a difference quotient of zero.
Clearly
(^) AO”* = 0 if ?i >m, A0” = n!.
If in 2-51 (2) we put = x^-^, = x, we have
A»” = a:Aa;'"-i + w"A\a;+co)'"-i
w
= X ^ ^ [w ^ _j_ A ^771-11
Putting a; = 0, we have the recurrence relation
/0\ ^ ^ W — 1
i-7 ^0^“^ + n j\
which in conjunction with (1) enables these numbers
successively. Thus
to be calculated
AO = 1,
AO^ = &jA0=co,
2-53] DIFFERENCE OPERATORS 37
A0^ = 2!,
A0» = coA0“ = 0)3,
u) a>
A03 = 2a)A0H2A02 = Ga),
A03 = 3!,
and so on. Expressions for these numbers will be obtained in
Chapter VI in terms of Bernoulli's numbers.
2*54. Expression of Difference Quotients in terms of
Derivates. By Herschel’s Theorem, 241, we have
{e-t- 1)« = f(E“-l)«0+^ (E“-1)”03 + ... .
Since E"-l=r^AjWe obtain
C^-n(e„e_l)n ^^o+,f,A02+... ,
where
AO» = [Aajlr=o,
which is equal to zero if s < w, and to w ! if s = w.
Thus
/n+1 n /n+2 n
CO-"(e“‘-l)’‘ = «" + /-rXTr. A0”+l + 7r-FAT,A0'‘+3+... .
Now from 2-5 we have
A = «-”(e“-0-l)'\
Thus
A0”+^
A0«+3
[See also 7*05.]
2'6. The Summation Operator P“i. If oj be a positive
integer variable, capable of taking the values 0, 1, 2, 3, ..., we
write
x-l
P (^) ~ ^x~X ”!■ '^a;-2 '^x-Z ~b • • • + *^^0 = ^ ,
(1)
38
DIPFEREN'CE OPERATORS
[2-6
where the notation p-^ is introduced for formal reasons f and
indicates the inverse nature of the operation. Indeed it follows
at once, if Uf be independent of x, that
APw P(all)M« - P^Uf
= ('U^ + Uig-i+ ... + Uq) ~ ('*^*-1 + Wa,_2 + ... + Mo)
so that the operator A neutralises the operator p-i, which is in this
sense an operation inverse to A- With the above notation we have
for example, ’
Pt.n-m+D'^t+m = “« + %-!+••• + Pn + 1 - P('^Wj.
When there is no risk of ambiguity we may conveniently write
P~^ w* = u^i + m*_2 + . . . + Mp.
These notations may be compared with
rx
u^dt and u^dx.
Jo Jo
We have at once from (1),
P(n«« = Mo. P(oN« = 0.
If, by affixing an asterisk, we now define a function u* bv the
properties ^
if 27 ^ 0, = 0 if a? <; 0,
we have
= (E-^+ E"H ... + E"^~H ...)«*
The operation can be repeated any number of times, thus
Pw «*= Pw [(E = (E
= ^x-z+2u^s + 3u^_^+... + (x-1)uo,
and this result is seen to be in agreement with (1) and (3).
r L. M. Milae-Thoiason, Proa. Oarrd>. PUl. Soc., xxvii (1931), 26-36.
2-6]
DIFFERENCE OPERATORS
39
More generally we have
P(x^ = (E “ 1)“^ 2 + •••
(6) = P(;l:„+1)
which expresses n successive operations with P”^ in terms of a
single operation. Also, if he independent of x, we have
(7) APf.r = APw [P(7)”+' = P(7r''
2*61. Theorem. JJ f{^) be a function of the positive integral
variable x such that A /(^) = '^x ? 6e independent of x, then
L Jq
We have
A {P w -/{*)} = M* - «x = 0.
Hence
Pw = constant = -/(O) ,
since P(0)^ = 0. This is the required theorem.
2*62. The following table exhibits the relations between the sums
and the functional values Uq, u^, ... .
P-^Ml
A Mo
0
p-^Wj
Ml
A Mo
3
P"®«3
P~^M2
A Ml
2
A Mo
P-^MJ
AMi
P“®W4
P-^«3
A M2
2
a
A Ml
p-^u^
Ms
A M2
3
P-^MJ
P-^M4
A Ms
2
A Ms
p-^u.
M4
A %
P-®«6
p-^Wg
P-^%
Ms
AM4
P-^«6
40
DIFFERENCE OPERATORS
[2-62
Each sum is formed by adding the members of the column on the
right, beginning with the member immediately above the required
sum. Thus
We note also that each column can be formed by differencing the
column immediately to the left. It will also be noticed that if we
change the origin, that is, label another entry with the sujSEix 0, the
resulting sum table will have each of its members altered in value
while the differences will be unaffected. Lastly, we note that all
entries labelled with the same suffix lie on a diagonal line.
Obviously in analogy with central differences we could also form
central sums ’’ by a mere change of notation.
2*63. Moments. Given a set of x tabular values, say,
Uq, U2, ^x-l}
their nth moment * about the point a; - 1 is defined by
sc-l
(1) M„=J^{t-x+l)”Ut= p^-l{t-x + l)'^u,.
If we express (x-t-l)”’ in factorials by tbe method of 2-12 we
obtain
^=() \ b /
where
(2) c» = A0",
a difference of zero (see 2*53). ' Thus
x-l n
i=0 8=0
n x-~s-l
Sv-A X—t—1
Zj
8=0 t=0
Since \ vanishes when t ^ x — s —1.
* See for example W. Palin Elderton, Frequency Curves ayid Correlation^
(London, 1927), chap. iii.
2-63]
DIFFEREN*CE OPERATORS
41
Hence from 2*6 (6) we have
(3) ( - 1)”M„ = ^ C, Ui='^c, p-»-i .
s~Q fi==0
Using the differences of zero given in 2-53 and noting that Cq = 0,
we have
My= - P-^M^,
M^= p-^u^+2p-^u^,
M^= -p-^u,-Qp-^u,-Qp-^u,,
and so on. The terms P~‘u^ (which are also called factorial
moments) can be obtained directly from the sum table of 2-62.
If the moments be required about another point, say y, we have
X-1
t=0
^=:0
which expresses the moment Mn, y in terms of the moments .
2*64. Partial Summation. We have
u^A'Vx = - ■»®+i A ■
Operating with we obtain
Pw A - Pw A •
Example. To calculate P^)Xa^,
We have /X a^= (a - 1) a®, and hence
PMxa‘‘{a-l) = \xa^ -Pwa‘'+^.
L. Jn
Now
P(-;/a*+i = a’‘+o«-i+...+a = ,
so that
P(„Ja;a*
w a” - a.
(a- 1)2 ■
42
BIFFEREl^-CE OPERATOBS
[2-64
The analogy of the formula for partial summation with the
formula for integration by parts should be noted. In fact if^
taking to be an integer, we make the extended definition
P(n)i ^ + • • • + J
the formula for summation by parts becomes
J P (n)ai A ^a: j
wticli wken co — > 0 becomes
2*7. The Summation of Finite Series. If we denote by
tbe xth term of a series, tbe sum of the first n terms is
P(ft) ^x+l ” +
To evaluate this sum we see, from tbe theorem of 2-61, that it is
sufficient to find a function /(tc) such that
A /(a;) = M*+i.
The general problem of solving this equation constitutes the
summation problem, which will be treated in Chapter VIII. For
the present purpose we require only a particular solution, and we
■shall now shew how such a solution can be obtained for certain
special forms of .
2*71. Factorial Expressions of the form By211(5)
Thus, by 2*61,
Tor example.
L m+l J(, m+1
1 . 2 .3+2 . 3 . 4+... + (re-2)(n-l) w = l{n-2){n-l)n{n+l).
Also
m(m+l)(m + 2) + (m+l)(m + 2)(m+3) + ... + (M-2)(w-l)n
= P(-/(a;+l)(3) - P(-Vi)(a:+l)W
= i(n-2)(n-l)n(n+l)-^(m-l)m(m+l){m + 2).
2-71] DIFFERENCE OPERATORS
Again, from 2*11 (10), we have
43
whence we get
‘ a(m+l)
= (ax + bY^\
in)
a(m + l)
Example. Sum to n terms the series
3.5.7 + 5,7.9 + 7.9.11 + ....
The icth term is (2x+5)(2x+3)(2cc+l) = (2a; + 5)<^> and the
required sum is therefore
'(ft)
(2x+7)(3) z=
L y
= M2w+7)(2M+5)(2n+3)(2n+l)-
J 01.
8 *
2*72. Polynomials. If the irth term be a polynomial in a;, we
could use the method of 2*71, having first expressed in factorials
by the method of 2-12. But from 2*12 (2) we have
4>{n) = ^(0) + nA^4(0)+?i^) A^i(0) + ... .
Putting (f){n) — P(n)^Wa.^.i, we obtain
^(0)==o, A^(0) = A^i,.-.,
so that we get the formula
Pin) ~ A — gr A ^1+ • • ■ •
Since u^. is a polynomial, the terms on the right vanish after a
finite number of differences have been formed.
Example (i). Bind the sum of n terms of the series
12 + 22+32+.. . .
2
Here = a;2, A^a: = 2a;+l, A^a = 2 and the required sum is
, 3n(n-l) 9^(9^-l)(n-2) _ n(n+l)(29^ + l)
n+ ~ + g .
44 DIPFEREN-CE OPERATORS [2-72
Example (ii). Find the sum of n terms of the series whose nth
term is n^+7n.
We form the following table of dilFerences :
«1
U2
Ms
M4
8
22
48
92
14
12
26
18
44
6
Hence the required sum is
I , 12n(n-l)(n - 2) 6w(m - 1) (n - 2) (n - 3)
2 ■ 6 ^ 24
=in(re + l){n®+n+14).
Another method of summing series of this type by means of the
Polynomials of Bernoulli will be explained in 6-501.
2-73. Factorial Expressions of the form xf-"™).
2-11 (7) and (9), we have
2;(-in+l)
A TT = 2^“”^ ■
-m+l
(aj-f l)(a!-f 2) ... (x+m)’
. (ax+bY~^-^^^
From
Thus
[a (a; -f 1 1 -f 6] [a (a; + 2) + 6 j . . . [a (a; + ti^T] '
Pw(a:+ !)(-’”> =
L- -a(w-l) Jj'
Example. Find the sum of n terms of the series
_i._ , 1 , 1
i-4.7'^4.7.io'^n:oj+- •
Here the xth. term is
1
2-73]
DIFFERENCE OPERATORS
45
Thus the required sum is
P(-;/[3(a;+l)-5](-3) = ^(3 . 1 -5)(-2)-H3(n+ 1)-5]<-2)
1 1
24 6(3w+1)(3m+4)’
These results are analogous to the formula
{"(ax + 61 - dx -
J^(axHr6) dx- a{m+l)‘
2*74. A certain type of Rational Function. If
where ax +b
and if (f>{x) be a polynomial of degree lower by at least two unities
than the degree of the denominator, then we can sum the series to
n terms. We begin by expressing <f>{x) in the form
<j>{x) = ao + «l'^'’x^-a2■y*u*+l+...+a„_l^)xVx+l^>x+2■••'yx+m-2•
This can be done by an obvious extension of the methods of 2-12,
or indeed by equating coefficients. It then follows that
so that the sum can be obtained by the method of 2*73.
Again, supposing the numerator of a rational fraction to be of
degree less by at least two unities than the degree of the denomi-
nator but intermediate factors alone to be wanting in the denomi-
nator to give it the factorial character described above, then, these
factors being supplied to both numerator and denominator, we can
obtain the sum. Thus, for example,
^£C+2 '^SC+S
^«^a+l'^a5+2'^ar-{-3
Example. Find the sum of n terms of the series
i.3.4'^2.4.'5'^3.5.6'
46
DIFFEBENCE OPEBATORS
[2*74
Here
x+1 _
" x{x+2){x+3) " x{x+l){x+2){x-\-Zy
= (a;+2)(-2) + (a;+l)(-3> + a;<-«
Pw “»+i = - (»+2)(-« - i (w+ 1)(->*> - 1 n(-3
_1 , 1 , 1 1
“3 12 18 n+3 2(w+2)(to+3) 3(w+1)(w+2)(m+3) ’
2’75. The form a^(f>{x}, ^(x) a Polynomial. We have
Aa“i)» = a®(«-l+aA)«'» = “®(«-l)(l + ^A) ‘Vx,
where 6 = a/(a-l).
If ^{x) be a polynomial of degree v, put
Vx = ^{^-bA+b^k-- + {-irb^A]<f>i^)-
Then
j\a’‘v„= a*{l + (-l)‘'h‘'+^ A
We have therefore
(1) p^J;a-^^j>{x+l)
fin-rl 2 V
= -ji [1 - & A+&' A- - + ( - 1)- A] <^(n+ 1)
-„^[i-6A+&'A-- + (-i)‘&’'A]<^(i).
Example. Find the sum to n terms of the series
12. 2+22. 22+32. 23+42. 2«+... .
Here «, = 2® and the required sum is
2«+i{l _ 2A + 4A} (m+ 1)2 - 2 {1 - 2A + 4A} 12
= 2»+H(n+l)2-2(2n+3)+8}-2{l-6 + 8},
= 2»+i(n2-2n+3)-6.
2-76. The form v^<f>{x), <f>{x) a Polynomial. Let ^{x) he
a polynomial of degree v. Consider the expression
f(x) = (p-^v^)<f>{x-l) - (p-2 v^) A<f>{x-2)
2
+ (P-® ®x) A 3) - ... + ( - 1)- (P—I A (a; - V - 1).
2-76]
DIFFERENCE OPERATORS
47
Since A («» Q = Wi A «* + s* A ,
we bave
A/(a:) = 'o^<i>{x) + {P-'^v^)
+ ( - 1)-- [( p- A - v) + (P--1 1,^) aV (a: - V - 1)]
= v^,f>{x) + {-iy ( p-‘-i v^) "a <j>(x-v-l)=v^<j> (x),
v+1
since A v- 1) = 0.
Thus, by 2-61,
(1) P(l) + = { P(l) ^x+i) ^ (^) - ( P(n)V:,+i) A ^ - 1)
+ (P(«f^x+i) A^(^-"2)- ... ,
Since P(n/ = 0 when n = 0.
This result enables us to sum the series whose xth term is v^(l>(x),
where ^ (x) is a polynomial of degree v, provided that we can form
the repeated sums
P(n)'^£C4-l> 5 = 1, 2, 3, ... , v + l.
Example. Sum to n terms the series whose nth term is
(n-c) sin {2an4-6).
Here <j>{x) ^ x-c, = sm.{2ax+b), and the required sum is
P(«)^ (ic + 1 - c) sin (2ax + 2a + 6)
= (n-c) P(^)^sm(2aa;+2a+&)- P(;;fsin(2a£i?+2a+i).
Now* A sin (2aa; + 6) = sin {2ax + 2a + 6) - sin {2ax + 6)
= 2sinasinf 2aa5+6H-a + ^y .
*It is interesting to note that, if we form the difference table for
sin(aa; + 6) or more generally for the terms in the same horizontal
line are in Geometrical Progression. This fact was employed by Briggs
and by Gregory.
48
DIFFERENCE OPERATORS
[2-76
Hus
P(;?sin(2aa;H-2a + 6) = ^ • ?— ( sin r2aa:+2a + &- a -
M SlIX -JQ
= j^sin (2an + b+a-^ + QOB{a + h)^ ,
P(;fsin(2aa; + 2a + 6) = +
” ncos{a + b)
2 sin a
Hence, after reduction, the required sum is
-csiana sin (na+a+b) n cos {2an + a + 6) sin na cos (an + b)
sin a 2 sin a 2 sin%
The repeated sums required for the method of this section can
always be formed for the t3rpes (ax+b)^^\ 4>{^), a^(j>(x), where <l>(x)
is a polynomial, since the operation with P”^ in each case leads to a
function of the same form. Repeated sums for {ax + b)^^^'> can be
formed, provided that the number of repetitions be not great enough
to lead to the necessity of evaluating + for which no
compact form exists in terms of the elementary functions here con-
sidered.
2*77. When the nth term of a series proposed for summation
cannot be referred to any of the preceding forms it is often possible
to conjecture the form of the sum from a general knowledge of the
effects of the operator and hence to determine the sum by trial.
For example, if (f>{x) be a rational function, then
i!^a^<f>(x) = a®
where is likewise a rational function. Similarly
A tan-^ <l> (x) = tan*”^ ^ (x),
where ^(cc) is rational if <f>{x) be rational.
Example. To sum to n terms, when possible, the series
P . X . X2 32 . X3
2.3 3.4 ■^"4.5
The rcth term is given by
2*77]
DIFFERENCE OPERATORS
40
Here we should evidently assume that
P(n) ^0^+1= constant.
Operating with Aj we have
(n + l)^X^+^ _ 1 ({a(n-{-l) + 6) X _ an +-61
(n + 2)(n + 3) ~~ n4-3 n + 2 j
Equating coefficients, we must have
(X-l)a = 1, 3a(X~l) + 6(X~l) = 2, 2(a + fc)X-36 = 1,
From the first two, (X-l)(a-f 6) = 0, whence from the third
h=z so that a = X = 4.
Thus the series can be summed if X = 4, and we have for the sum
r(x~i)4*+n«_ 2
_ 3{x+2) Jo“ 3(w + 2) ■*'3‘
This example is due to Boole, who explains the peculiarity as
follows :
= X"-
4X^ X^
^ + 2*^^+ 1 ’
\n
SO that unless X = 4, in which case the term :r destroys the corre-
n+l
-4X^
spending term ^n-i> we should require the sum of a series
whose nth term is
_X"
n + 1*
Such a sum cannot be obtained in terms
of the elementary functions considered here (but see Chapter IX).
EXAMPLES II
1. Prove that
(i) A log U (x) = log (l + ;
u
(ii) A log (% %_! . . . Mx-m+l) = log
^ar-m+l
2. Prove that
Asin(^J^-f-6) = (2 sin^a)^sin{a3J+6-f|n(a+Tc) } ,
n
Acos(ax + &) = (2sm|a)” cos{aa;-f 6H--|n(a4-7r) } .
n
Obtain corresponding results for the operator A? deduce the
results for the operator D^. “
50
BIFEEREJSrCE OPERATORS
[ex, n
3. Prove that
tan ax =
sin a
cosaa3Cosa(aj-i- 1) '
A tan-'^ ax = tan”^ ^
4. Evaluate
5. Find the first differences of
2*smp, taii|;, cot (a. 2*).
6. Shew that
A™ =
<u 'Oo^
7. Prove that
8. If cj) (X) be a polynomial, shew that
9^(E)0-=Ef(E)0-\
and deduce 2-53 (2).
9. From HerscheFs Theorem, or otherwise, deduce the secondary
form of Maclaurin's Theorem, namely,
^(«) = ^(0) + ^(D)0+^^6(D)02+^'^(i3)03+... ,
where ^4(D)0* is the value when t = 0 oi (f>{D)t>.
10. If E”0® denote n®, prove that
^(E«) 0“ = re®^(E)0®.
11. Shew that the differences of zero
AO”, A0"+^ A0”+^ ...
form a recurring series and find the scale of relation.
12. If C” = i A 0“, shew that
ib !
Cl = Clz\+n(Jl_-^.
DIFFERENCE OPERATORS
51
EX. Il]
13. Shew that
Uq^u^x+-~ +... = e® + ^ A'^^'o + 2! A^o+--- | ,
where is a polynomial.
14. If Sn = T“ +07 ^ . + ... + -~“-v , shew that, if m > 2,
1 . 71 2{n-~l) n . 1
(SiA-^2A+...)o- = o.
71
15. Express A ^ series of terms, proceeding by powers of x,
by means of the differences of zero.
Find a finite expression for the infinite series
l’” .a; - 3”*.^ + S”*. ,
where m is a positive integer. If m = 4, shew that the result is
{x - 6x^) cos x~{7x^- x^) sin x.
16. Prove that
f{x^){xE = {xE )”‘f(x^ + m)u^.
17. Find Un from the relation
^i”Un =
71-0
1 - 71 - 4«2
18. If t’^u„ = /(e‘), prove that
n~0
W„ =-^^^^0".
n\
19. Find a symbolical expression for the nth difference of the
product of any number of functions in terms of the differences of
the separate functions, and deduce Leibniz’ Theorem therefrom.
20. If the operator A on n alone, prove that
TjCl+l TT-
m being a positive integer greater than a and the even integer
next greater than a + 1.
52
DIFFERENCE OPERATORS
(KX. II
21. Shew that
"a*9. = AL"-'-i+ a'i”
n+i
22. Prove that
A 1*’+^ = (w+ i)A i^+n a'i”,
and apply the formula to constructing a table of cliffercncoa of
powers of unity up to the fifth power.
23. Prove that
( - l)«/(a:+ ww) = /(*) - 2 ( j) + 2= g) Vf(x) - .
24. Prove that
A V/(®) = Am
ui ti* 2«>
A V/(^) = A/(x).
25. Sum to n terms the following series
(i) 1.3. 5. 7+3. 5. 7. 9 + ....
(ii)
1.3. 5. 7^0. 7.9
+ ... .
(iii) 1.3.5.10+3.5.7.12 + 5.7.9.14 + ....
(iv)
_10 , 12 , 14
1.3.5 077 ■^079'^
(v) 1.3.5cos0+3.5.7cos2e+5.7.9cos3O + ... .
(vi) l + 2acos0+3a2cos20 + 4o®co8 30 +
26. The successive orders of figurate numbers arc defined by
tbs; that the ajth term of any order is equal to the sum of
nAhfft ! preceding, while the terms
of the &st order are each equal to unity. Shew that the xth ti>rm
01 tile nth order is
EX, ll]
DIFFERENCE OPERATORS
53
27. Prove that where /(./) is equal
to the expression
. m ...v - . m .
sin (2x "> 1 ) cos {2x)
• /*> -t %
sm-^(2a;+l)
T"
cos-H-(2a:+2) sin ^ (2a; + 3)
/„ A-?i(3;) + ... .
(2sm^j 2sm2j
28. Prove that
Pw
= (JlV” + (^“i)'A^(a=)-...}
4-CQ + (7jir'f...4-
and determine the constants Cq, Cj, ... ,
29. Use the result of Ex. 28 to discuss the summation of the
'^1 - ^2 + % *^4 + * • •
to n terms. Consider the forms of given in sections 2-71-2-70.
30. Prove that to n terms
(i) — -v; H~ + -t — + . . . = cot ^0 — cot 2”"'^0 ;
^ ' sin 0 sin 20 sm 40 ^
.... ^ ^ _ 2sinn0
cos 6 cos 2 0 cos 20 cos 30 “ "" cos {n + 1)0 sin 20 *
31, Shew that cot'‘^(y + 5f + ra;2) can be evaluated in finite
terms if == 4(j)r-~ 1). Calculate :
P<») iTT7:^-i iT^ ’ P(«)
1 log tan 2“a 2® (x - 1)
) 2^ ’ i^i+iy
rw l + a:(a;-l)X>‘’ 2* ’
32. It, is always possible to assign such real or imaginary values
to s that P~’-/(a5) can be evaluated in jSinite terms, where
f( ) — («+P^+Y^^+--- + '^^")
a, j3> , V being any constants, and u^. = ax +b, (Herschel.
54
DIFFERENCE OPERATORS
fMX-. II
33. Shew that
M„+% cos 26 + M2 cos 40-1- ...
= — -
4 sin^0 8 sin^0
4^sine + ,-A-®w
16 sin‘'0
cos 20 •
32 sitr'’’0 '
34. If (^(x) = Vg + ViX+v.,x~+... , yhew that
Mq«0 + “1 V + + • • •
= Mo,^(a;) + 3:f (a:)AMo + ^|f'WA'«u I -- •
and if ^ (x) = ■!;(, -t- VjX -h H- . . . , then
UgVg + UjVlX 4- +•••
= Ug(l>{x)+x^(j>{x-l) . i^Ug+Q^(l,(x-2) . ^Ug \ ... .
((iiideriiiaiui.)
35. If Sn — and p = ?«.(■;« t- 1), shew that
Sn = P^f(p) or (2?jn-l)y)/(;ii),
according as n is odd or even,/(j)) being a jiolynomial.
36. Prove that the number of ways in which an integer which i.s
the product of m prime numbers can be e.xpressed us a product of
factors relatively prime to each other is
m r
r=0
Prove also that 8^ satisfies the recurrence relation
37. Prove that
and that, if m is a prime number, - 2 is divisible by tn.
CHAPTER III
INTERPOLATION
In the practical applications of the finite calculus the problem of
interpolation is the following : given the values of a function for a
finite set of arguments, to determine the value of the function for
some intermediate argument.
In the absence of further knowledge as to the nature of the
function this problem is, in the general case, indeterminate, since
the values for arguments other than those given can obviously be
assigned arbitrarily.
If, however, certain analytic properties of the function be given,
it is often possible to assign limits to the error committed in calcu-
lating the function from values given for a limited set of arguments.
For example, when the function is known to be representable by a
polynomial of degree n, the value for any argument is completely
deterininate when the values for n+ 1 distinct arguments are given.
In the present chapter we propose to obtain certain formulae based
on the successive differences of the function for the given arguments
and to investigate the remainder term, the knowledge of which will
enable us to decide as to what further information is necessary to
ascertain limits within which the interpolated value represents the
value sought. In actual calculations there is, of course, another
source of error due to the fact that the known values are usually
approximations obtained by curtailing at, let us say, the fifth
figure a number which contains more than five figures. For an
investigation of this error see papers by W. F. Sheppard.*
The basis of the interpolation formulae about to be obtained is the
general formula of Newton for interpolation with divided differences.
* Proc, London Math. Soc. (2), 4 (1907), p. 320 ; 10 (1912), p. 139.
55
INTERPOLATION
56
[a-0
This formula with its remainder term has aln^atly been gi\'eiL The
formulae of Gauss, Stirling and Bessid were knovvui to Ni'wton, and
if for brevity we do not attach his name to them, it dtH\s not detract
from his credit in discovering them.
3*0. Divided Differences for Equidistant Arguments.
If in the formula T3 (1), we put
5 = 1,2, 3, ... , /a
we obtain for the divided difference the expression
/(a?+ w)
n\ CO”
/(a;4-(^-l)co) /(:r i {n 2) c.>)
(^-1)! ifco” ■(//■■ 2)! 2! c.>“ ■■
= ^ A/(i^) from 2-f) (2).
Since the arguments in a divided difference can written in any
order we have thus proved the following theorem.
If the arguments x^, X2, taken in a certain order form an
arithmetical progression whose first term is x, and whose amimon
difference is <0, the divided difference of f{x) formed with these argu-
ments is given by the relations
(1) [W3 ... a;„ J = i A/(a;,) - A'‘fO.)-
Again by 1-2 (2), we have
K^2-^n+i] = 2 /(«)(5),
where 5 lies in the interval (x^, x,+no).
Thus we have
(2) Af(x.} =/<">(?),
ui
where ? is some point of the interval (»„ x, +nw).
In the notation of differences this result can be written, usini;
2-1 (5), ^
(3) /(«)(^) =
Ms result shows that the Mth column of differences formed from
a table of functional values for equidistant arguments places before
us a specimen set of values of the nth derivate of the function, each
INTERPOLATION
3«0]
57
such derivate being multiplied by which is a constant for the
column in question.
3*1. Newton’s Interpolation Formula (Forward Differ-
ences). Consider the following table of functional values and
differences.
Argument
Function
a
/(“)
Alia)
(X + CO
/(a+w)
Zl/(a + «)
A^fia)
A^fia)
a + 2(0
f{a + 2(j>)
Af{a + 2a)
ZlV(« + «)
Z|3/(a + «)
a + 3co
/(a + 3w)
JV(« + 2co)
If in Newton’s general interpolation formula with divided differ-
ences, 1-1 (1), we write
X, — a~h(s-l)oi, 5 = 1, 2, 3, ... , w,
we have by 3*0 and 1-2 (3)
[^1^2 - a:,+i] = ^ /!“/{«).
[OTj ... x„] = ,
so that the formula gives
(1) /(^) = f{a) + {x-a) <o-M/(«) + ~ W + -
+
(x-a){x-a-oi) ... (a; - a - nco -f 2co)
W"+M"~V(«) + (»)>
where
(2) R.ix) = /«(E),
i
and ^ lies somewhere in the interval bounded by the greatest and
least of X, a, a + nca - CO.
This is Newton’s Interpolation formula with forward differences.
The diJBEerences employed with this formula lie on a line sloping
downwards from / (a). The formula gives the value of f{x) in
58
INTERPOLATION
[•M
terms of f{a) and the differences of /(a) provi(i(‘«i I ha,t \v<‘ can
calculate the remainder term Tlu‘. formula assuiucs a
simpler form if we introduce the fhase 'p, wlua‘(‘
(3) p=:{x-a)jo},
which represents the ratio of the distance betwi'cn t he '' ])oint s ”
X and a to the tabular interval of the argument. We t hen ohlaiii
(4) /(iK)=/(a)+j)/j/(a) + (|jaV(0 + (3j/1VV') i
which is the most convenient form of Newton’s formula, wit li forward
differences, the value of p being given l)y (r>).
If in (4) we omit the remainder term, we ol)tain Newtoirs lnt(‘r~
polation polynomial (see 1-9)
(5) /«-!(») =/(ffl)+pa/(«)+(2)av(a)'i ij-'i” ‘/(«).
which assumes the values of/(a;) at the points
a, a + co, ... , I) cr).
It follows that neglect of the remainder term is equivahait to
replacing /(a?) by this interpolation polynomial. drgrtM‘ of
approximation attained by this process of polynomial itdorpolat ion
of course depends on the magnitude of the m^glecdcd r<*ma,in<li‘r
term. This will be discussed in section 3-12.
We may here observe that the interpolation polynomial (d) eati he
written symbolically in the form
(6) ^»-iW = (i+a)f„ ..,)/{«),
where the su&x 7i~\ indicates thut the cxpcinsioii of t<ht* operiif or
by the binomial theorem is to cease after th(j term in , | ‘ bus hoen
obtained.
Newton s formula, or rather the series which arises from it when
thennmher oftermsisnnhmited, is of great theon'tical importance,
as will be seen in Chapter X. For practical mimerical interpolation
the central difference formulae to be obtained latter are preferred.
Near the beginning of a table, however, when (jcntral differeticeH are
not given, Newton’s forward formula is available.
3-1]
INTERPOLATION
59
In the above work we have written ^ewtoii’s formula in the
notation of differences, the form best suited to numerical applica-
tions. We can, however, use difference quotients. Using 2*1 (5),
we see at once that (1) can be written
(7) fi^) + + A /(«) + ■••
iU ^ * oj
{x-a){x-a-i>i) ...(a:-a-ww + 2c5) r, ^ , -o , \
+ ' IV-IT! ^ A/(a) + -K«(a:).
Or in the factorial notation,
(8) / {x) = f{a) -I- (x - a) A /(a) + A /(«) + -
When 0) -> 0 we obtaiii Taylor’s Theorem, namely,
fix) = f{a) + ix- a)f' (a) + f" {a) f .. .
(H-1)!
where ^ lies in the interval {a, x).
The formula of this section is often referred to as the Gregory-
Newton formula, since it was actually discovered by James Gregory
in 1670.*
3*11. Newton's Interpolation Formula (Backward
Differences). Here we consider the table
a -See
/(a-3co)
a - 2co
/(a-2m)
a-6)
fia-oi)
a
/(a)
J/(a-3o))
J/(a-2co)
zlV(a-4w)
/f^f{a-Z(^)
A^fia-2a)
A^fia-^)
ZlV(«-3o>)
* The actual MS. letter from Gregory to Collins which gives this formula is
dated 23 November 1670, and is preserved among other of its contemporary
documents in the library of the Royal Society.
60
INTERPOLATION
If in M (1) we write
= a-{s~ l)co, 5^1, 2, ... , n,
we have by the theorem of 3*0
[^1^2 * • • ^5+l] ~ ^ [ A^f(^ •
Thus we have
[:mi
f(x) =/(a) + (a;-a)«-M/(ffl-<o)
(x-a) (x-g + o)
2!
o>~^A~f{a - 2a>) f
or introducing again the phase p = (x-a)l u, this Ix'comea
/(s;) =/(fl)+i)Zl/(a-«)+^^!-i^ J“/(a--2w) f ... ,
which can be written
(1) /(®) =/W+^’Zl/(a-w) + ^^2 + ...
+ w - 1 + C' ’* * j (?),
where | lies in the interval bounded by the greatest and least of
X, a, a - (w - l)u. This result could also be obtained by writing the
tabular values in the reversed order, difierencing, and’ tiien apply-
mg the forward formula.
The differences employed with this formula lie on a line sloping
upwards from f {a). The corresponding interpolation polynomial
obtamed by omittmg the remainder term may be written syml)oli-
cally in the form
d'n-iCa:) = +
/(«)
where the suffix again indicates the index of the last term of the
bmomial expansion which is to be retained.
nnkt^^ ^ fcaoiward formula has its practical application to inter-
pdation near the end of a table, when central differences are not
given.
3-12]
INTERPOLATION
61
3*12. The Remainder Term. The process of interpolation
applied to the values in a given table cannot of course give an
accuracy greater than that of the values in the table, which are in
themselves usually approximations. In attempting to attain the
utmost accuracy which the table permits, when a given interpolation
formula is used, it is common practice to omit from the interpolation
formula the first term which ceases to influence the result obtained.
The question then arises as to how far the result so obtained repre-
sents the desired approximation.
The error in the approximation arises from two sources : (i)
errors of rounding, inherent in the tabular matter and the subse-
quent calculations ; (ii) errors due to neglect of the remainder term.
With regard to errors of the first category we shall content ourselves
with the observation that, in so far as they arise from subsequent
calculations, these errors can be minimised by using one or two extra
figures which are subsequently discarded. As to the errors arising
from (ii) we shall make some observations, with particular reference
to Newton’s forward formula, but which are of general application.
(а) In numerical work we naturally take x between a and a + co,
so that the phase p is positive and less than unity. Consequently
and opposite sign.
(б) If we can conveniently calculate (a;) we can generally state
upper and lower bounds to the value of this derivate in the interval
(a, l)<d) and thus delimit the error due to neglect of the
remainder term.
(c) If {x) have a fixed sign in the interval {a, a+ (n - 1) co) and
f(n-¥i) (x) have the same fixed sign in the interval (a, a-h-nco), then the
inclusion of an extra term in the interpolation formula gives
Since, by (a), and opposite signs, so also have
lt„(x) and ; and consequently
Unix) (j)zl"/(«)
02 INTEBPOLATION i;i-12
that is, R^{x) is less than the first term omitted from the formula and
has the same sign. This result is called l)y St(‘iren.s.‘n the. Error
Test.* The test depends essentially on and A’„n(:r) having
opposite signs.
(d) If nothing be known about the value or sign of/'">(^'). «'(> can
only regard the results of interpolation as a working hypotlie.sis.
This in particular would be the case if the tabular malt.er were
empirical. In such cases we might bo incliiu’-d to (“stimate flu? value
o£ fM(x), on the grormds of the last part of section .'J-O, by an
examination of the nth column of differences. That smdi Gonj(“cture
may be fallacious is seen from the following table ;
We have
X
0
1
2
3
0
1
4
9
V -fi
/(0-5) = 0 + -5xl--^ x2
■5 X -5x1 -5
G
4*
= 0-25 + ,V
Tte third difference is zero, so that an estimate of tin*, error term
would be zero and we would conclude that /(0-5) ()*2r>.
This is correct iif{x) = x^. If, however,
f(x) = x^ + sin Tzx, (x) = “ 7V^ cos nx ,
the maximum value of which is tt®, and the actual error is I .
It might be contended that the instance is extremcily art illeial
To this we answer that a satisfactory mathematical th(K>ry auist not
exclude possibilities of such a nature, and, secondly, that if talmlar
matter be collected from observations .made at equal intervids (say
of time), a periodic term might quite well be masked in this mmimr.
* J. F. SteiBfensen, Interpolation^ London {1927).
INTElil^OLATlON
63
3*12]
Example, From the following values * calculate
sm0*lG04, cos 0*1616.
X
sin X
A
cos X
A
0*160
0-15931 820G6
9871475
0-98722 72834
1598118
*161
■16030 53511
1604
•98706 74716
9871
9869871
1607989
162
•16129 23112
•98690 66727
Using Newton’s forward formula, we have for the sine
The coefliclent of the second dijfference is |x*4x -*6= -*12,
while, since o = •{)()], the coefficient of the remainder term is
X *4 X - •() X - 1*6 X (-001)^ = 6*4 x IQ-^i.
Since -cosx, the remainder term contributes
- 6*1 X 10-11 X *99,
that is, 6 in the eleventh decimal place. We have then, treating the
tabular values as integers,
sin-1604 = 1593182066 + 3948590*04- 192*5- *6 = 0*1597130848.
For the cosine, using Newton’s backward formula, we have^ = ~ -4
and the coefficients ~ *12, - 6*4 x 10~ii.
Here = sinx, so that the remainder term contributes
- 6*4 X 10-11 X *16= - lxlO"ii.
Thus •
cos *1616 = 9869066727 + 643195*6+ 1184*5 -*1 = 0*9869711107.
In these values the only errors which can be present are those due
to rounding.
3*2. The Interpolation Formulae of Gauss. These are
obtained from Newton’s general divided difference formula, 1*1 (1),
by means of a special distribution of the arguments
* C. E. van Orstrand, Nat, Academy of Sciences, xiv, (1921), Part 5.
64
INTERPOLATION
It is again convenient to introduce the phase
(1) j) = (x-a)lo>,
and to write
fix) =/(a-f^?6>) = u^.
With the central difference notation of 2-0 1 we tlien have the
table
(2)
a + co %
If we put
(3) a?! = a, X2, = a + 5co, a'2,,4,1 - (i-
the theorem of 3-0 gives
^•“2s
[a^iTg . . . Kjs+i] = j.2jy| ~ ~ ( 2,v ) ! '
while
(X-Xj) {X-X2)...{X-X2,)
= i^- %-i) (» - a:2s_3) ...{x-X3){x-Xi){x- a-,) {x - .r,j) ... (a: -
= (p + s-l)(p + s-2) ...{p+l)p{f~l){p-2) ... (p-.s)oj“’
Thus
(a:-Xi)(a;-X2) ... (x-x,Xx^x^...z^„,] = MS‘* w„,
and similarly we can shew that
(X-Xj)(x-X2) - ix>-X^^^)[x^X^... 07.2,, .J .-= (.£,J,y S’*"' Mj.
If then we make in 1-1 (1) the substitutions given I)y (3), we
obtain
f{x) = U^ = Uo+i^)^Ui+ (I) S^Mo -I- J ^ j §3 « J
'^'2] INTEHrOLATION O5
Ihis is Gauss forward formula, and is used in conjunction with
the zig-zag scheme of differences shewn in (2).
If n = 2m, we liave
^^2rn ““ - 2^2) • • • - ^*2 (?) / (2w) !
and Gauss’ forward formula becomes
(1) /(-G
v.’So*—
where ^ lies in the interval (a — - 1) w, a + mco), when x lies in this
interval.
If fi = 2m -4- 1, we have
(5, /w = - 1 i” -- *)
where 5 hes in the interval (a~ mw, a4- mca), when x lies in this
interval.
Gauss’ backward formula is used in conjunction with the table
a - 0)
(6) Sw„S 8^u_^
To obtain the formula we write
= a, = a- s<j>, = a + so>
in the formula 1-1 (1). We then obtain hj the method described
above
{x - Xi) (x - Xa) .. . (x - Xj.) = Mo ,
(x - Xj) (x - Xa) . . . (x - X25+1) = (is + 1) ’
66
so that
INTERPOL ATI OK
f{pc) — Wj, __ ^ | , ^/ ^ J ’^a
which is the required formula.
If n = 2m, we have
■y.;.
(7) /w = », = «.+ s'(0*“"'”-i ' K
-er)“=-/-=-(a
where a: lie in the interval (a-m), « i (wi |)fo). whih fo:
n = 2m+l,
where a: lie in the interval {a-mca, a + 7tm).
It shonld be noted that if in (5) and (8) we omit t he .vnuiinder
terms, the corresponding interpolation poIynomiiiLs terminate ;it Mie,
same difference and therefore both agrees with f{x) iit t he s'une
points, and consequently coincide. Thus (lauss’ forward f.^rmuhi ha.s
the same remainder term as the backward formula if t;h(' la.st, dilTer-
ence used m each be of the same even order, and both formulae give
tJie same result. ”
Again, since
V 2s y 26-
we see that, in-the forward formula the sign of p hi* (dumged, the
coefficients of the even order difterences coimude with the com-
in sim to +1, ^ differences are equal in niagnitmle but, oppo.site
m sign to the corresponding coefficients in the backward formula.
lOTERPOLATION
3-3]
67
3-3. Stirling’s Interpolation Formula. Stirling’s formula
is obtained by taking tlio arithmetic mean of Gauss’ forward and
backward formulae.
We have
i V 'i ^ ' .S') (p H- 3 - 1 )...(??+] ) f {f- 1 ) „ . (p - s)
2s i- i/ ' (2s f l)!
(2.V+1)!
(P ^ ’ i , ('P ^-•'>•'^_2p(p + s-l)...(p-s+l)
V 2s / \ 2s / “ (25)1
_ 2f (p2 - P) {f - 22) . . . (p2 - s - P)
“ (2s)l
Taking the arithmetic mean of 3-2 (5) and (8), we obtain
(1) m - «„ -
.f-o (2s+l)!
V p2(p2-P)(p^-22)... (p2_7ri2)
+
« I
where, as ht^fore,
(2.)!
p(p2„ p) _
(2/M-fl)!
Uq
S2»Mo
0)3“+l/(2™+l)(^),
p (:r - a) / co, Uj, =^f{a+jm) =f{x).
This is Stirling’s formula. The dilferences employed lie on a
horizontal line through thus :
(2) a Uq g.S'Wo ^"*^0 —
Stirling’s formula is completely symmetrical about = 0 and can
therefore be vrsed for either positive or negative values of p. In the
form (1), which terminates with a difference of even order, the re-
mainder term is the same as in the formulae of Gauss which termi-
nate at the same difference. Hence from the point of view of
numerical calculation the formula of Gauss is superior in that there
is no necessity to form mean differences.
By taking tlie mean of 3-2 (4) and (7) we can obtain the remainder
term of Stirling’s formula when the last difference used is of odd
order. It will be seen that this is not of a very simple form.
68
INTERPOL.ri’ION
[3%3
Stirling’s formula written in full for rn ™ 2 is
(3) f{x) = U„ = + +
+ “ii
where x lie in the interval (a 2o>, a-f- 2o)).
The corresponding interpolation })o]ynornial o!)taine(l hy omitting
the remainder term in (3) agrees with \-i) (•!), whi<di may he n‘ganiecl
as a generalisation of Stirling’s fornuila for unequal intervals of the
argument.
Example, Calculate exp (-0075) from tlui following : *
X e®
zl
A~
0-006 1-00601 80361
100652] 2
-007 1-00702 45573
10075282
10070
•008 1-00803 20855
p = -5.
The coejficient of the second difference is -125, and since 6> •001
the remainder term is - *0625 x (-OOlf 6'"^ - 6 x io approxi-
mately.
Thus, using Stirling’s formula, we have
exp (-0075) = 10070245573
+ l (10065212+ 10075282) f J (I(K)70) - -6
= 1*0075281955.
3-4. Bessel’s Interpolation Formula. BeHscl’s formula is
obtained by taking the arithmetic mean of Giuih.h’ forward formula
with initial argument a and the corresponding backward formula
with initial argument a + w. We choose the forms which terminate
with a difference of odd order, that is to say ;b2 (4) iimi (7).
* C. E. van Orstrand, loc, cit p. 63.
3-4]
INTERPOLATION
09
With 'p I io, w(‘. Iuiv(^
/(x) Uo-
.’v /
52-’+%j+ ^
fi 0 ^ ^.s* + 1 /
V fp-\-s-l
p \~ ni- I
The second of thes(.‘ has been obtained by writing p-l for p in
3-2 (7) since the initia] argument is here a-f co. The remainder term
is the same in both since each terminates with the same difference,
namely Taking the arithmetic mean, we obtain
where x are in tlu‘ interval (a~(m-])co, ft + mo)).
This is Bessel’s formula. There is symmetry about the argument
a + |a), for writing - p + i for p - i-, we have
fp + s-u^fs-p-.
so that the coefiicients of the differences of even order are unaltered,
while the other coefficients mendy change sign.
It is convenient to replace ^ + (jo - 1) by UQ+pSu^, The
first summation above is then from 5 = lt0 5 = m~l. Written in
full for five dififerences, we have
+ (j?+i)?^(p-|)(p-i)(y-2) g5
4- (P + 2)(p+l)p(p-l)(y-2)(y-3)
70
INTERPOLATION
[3-4
The differences used with Bessel’s formula are shewn in the
scheme
a Uq
The formula may be compared with 1-9 (3), which shows the more
general form for arguments which are not equidistant.
If the last difference used be of even order the remainder term is
not so simple.
Example. From the following table * of the Complete Elliptic
Integral K, find the value of K when m = 0-032 whore ni{ = /c^) is
the squared modulus.
m
E
A
A^
A^
A
0-01
1-5747 45562
3994351
•02
1-5787 39913
4040429
4G078
999
•03
1-5827 80342
'4087506
47077
1022
23
•04
1-5868 67848
4135605
48099
1064
42
•05
1-5910 03453
4184768
49163
•06
1-5951 88221
Using Bessel’s formula, the required value (p = -2) is
1682780342 + -2 x 4087506 - -08 x 47588
-f -008 X 1022 + remainder.
In the absence of a convenient formula for the fourth order
derivate we make the hypothesis that this is approximately
represented by gin^e ce = -01 we have for the remainder the
hypothetical value + IQ-® x -0144 x 33 = 5 x 10"“ Thus we obtain
K = 1-583594045.
* L. M. Milne-Thomson, Proc. London Math. Soc., (2), 33 (1932), p. 162.
INTERPOLATION
71
3-41]
3' 41. Modified Bessel’s Formula- Neglecting the remainder
term, BesseFs formula correct to differences of the sixth order can
be written in the form
j{x) = UQ+f 8ui + [J.82 Ui + [S* Mj - A; S® m j]
(p + l)p(p-l)(p-2)
4!
- 1 fjiS® u^] + A-\- B,
where
A =
pIIT
sf
_A+l(p+l)(?5-2)]s5Mi,
B
_ (y + l)y(ff-l)(y-2)
4!
[^+^0(?> + 2)(j5-3)][xS«Mj.
The mean value of A over the interval _p = 0 to is
and the mean value of B over the interval 0 to 1 is
1 rili 191
'2520 J'
These mean values vanish if we take
(J,8® Mj.
13
191
Putting
V “i = ~ iii “i’
we have the modified form of Bessel’s formula, namely,
m = ^„,+£(£r|(Ezi) v„,
+(?±2K£+liE(£^)^..„„
4!
which includes the effect of sixth order differences. The coefficients of
the differences in A and B in the interval 0 ^ ^ 1 are of the orders
0*00002 and 0*00003 respectively, so that, if S® % and do not
exceed 10,000, the maximum errors which would arise from the
72
INTERPOLATION
[3-41
neglect of A and B would not exceed 0*2, 0*3 units of the last digit
respectively. Actually we use rounded values so that the error may
be greater.
The above method of modified differences can of course be
extended to diff“erences of higher order and to other interpolation
formulae.
Example. Consider the following table* of where
^3(01 t) is the value of the theta function %(x\ t), when a; = 0,
arranged according to values of m, the squared moduhis.
m
V^(0|t)
A
A^
A’^
A*
A^
Zl«
0-70
0-76687 78205
54346488
1161883
68293
5889
788
109
•71
•76144 31717
55576664
1230176
74970
6677
933
145
•72
•74588 55053
1305146
7610
166
Forming the reduced differences we have the following
table for use with Bessehs modified formula.
m
V^OIt)
0-70
0-75687 78205
54346488
1196030
68208
6257
•71
•75144 31717
55576664
1267661
74869
7111
-72
•74588 65053
Calculating the function for m = *706, either by using all the
differences or by usii^ the modified formula, we get 0-7536313968.
3*5. Everett’s Interpolation Formula. This formula uses
even differences only on horizontal lines through Uq and as in
the scheme
a Wft
a + oy
* L. M. Milne-Thomson, loc. cit. p. 70.
3*5]
INTERPOLATION
73
Gauss’ forward formula ending with an odd difference can be
written
f{x) = Uo+j>Bui+
S
+
.•li
p + m-l
’Mn +
p+s
2s + l
§2s+l
2 m
^ C02™/(2«)(^).
The term in curled brackets is equal to
Now
_l) |(2s + l)S2*M„ + (p + s)S2»+%i} .
(25 + 1)!
(25+l)S2»Mo + (i3 + s)S^“+^Wi = (25+l)S2»M(, + (p + s)(S2“Ml-a2»Mo)
= (jJ + 5)S‘^‘'Ml-(p-S-l)S2sWo.
Hence we have one form of Everett’s formula, namely,
m = u,+pSu, + ’f
A more symmetrical way of writing the formula is obtained by
observing that
V 2s + l / ~ V 2s + l /’
Mo+J)Smj = Ug(l-p)+ptii.
Hence introducing the complementary phase p', where
p' = 1-p z= (a+o)-x) 1 00,
we have the symmetrical form
f[x] = pu, + (£+ u,
+P'^o + "t (t+ 0
where x lie in the interval (a - (m~ l)o), a+ wco).
Everett’s formula is useful when employing tables which provide
even differences only, a practice which saves space and printing cost
but which offers little advantage to the user of the tables.
74 INTERPOLATION [3-5
For numerical values of the coefficients in Everett’s formula as
well as the formula of Gauss the reader is referred to E. Chappell,
A Table to facilitate Interpolation by the Formulae of Gauss, Bessel
and Everett (1929). (Printed and published by the author, 41 West-
combe Park Road, London, S.E. 3.) The coefficients are given at
interval 0-001 for the phase p and for differences up to the sixth
order, and are so arranged that the coefficients in Everett’s formula
for p and the complementary phase p' each appear on the same page.
Another table (of Everett’s coefficients only) is that by A. J.
Thompson, Tracts for Computers No. F, 1921, Cambridge Univer-
sity Press. The latter book gives many numerical examples of
interpolation.
3*6. StefFensen’s Interpolation Formula. Gauss’ forward
formula ending with an even order difference, 3-2 (5), can be written
m
f{x) = Mo+ D
5 = 1
j- + R^m+l (^)
The term in brackets is equal to
1 +
2sl 2s-l ,
) |(j9 + s) -{p-s)
u
Now
so that we have
/p + s-1 -p + s
\ 25 25
II
o
1
which is Steffensen’s formula. The formula employs odd differences
only according to the scheme
a -CO u_
INTERPOLATION
75
3*7]
3*7. Interpolation without Differences. The problem of
interpolation without the use of differences is solved in principle
by Lagrange’s formula 14 (3), which gives
(1)
m
(x,)
(x-Xj)(x~Xo) ... (x-x,^).
From this we can obtain a formula equivalent to Gauss’ formula
by substituting the proper distribution of arguments. Thus to
obtain a formula completely equivalent to 3-2 (4), we put
^2s-i ~ a - {s ~l) CO, “ a + .vco, s ~ 1, 2, 3, ... , m.
Introducing the phase p = {x - a) j co, we have
<j) (x) = (p-h m "f m - 2) ... (p - m) co^
= (m-s)! (mH-s- 1) ! (-
^'(3?2s) = (m + 5-1)! (m~6-)! (-
so that (1) gives
f(x] = V (p + ”'t-I)...(p-m) jf(a- ,S(0 CO ) _ /(a + «o) 1
s=i (m + s - ] ) ! {m - s) ! ^ 1. ^5 + .s - 1 ji~~s j
+ remainder.
This can be written in the simpler form
X «) _ /(« + 1 + 1 ^ f(3™) (1)
t ^ + 5-1 p-s J V 2'm /
Other formulae of this nature can be obtained by varying the
distribution of the arguments. The practical objection to the use of
the Lagrangian formulae lies in the excessive labour of numerical
calculation involved. In using interpolation formulae founded on
differences the order of magnitude of the terms becomes progressively
less. Moreover, if it be found desirable to include further differences
it is only necessary to add more terms. In Lagrange’s formula every
term is of equal importance and when another functional value has
to be included the calculation must be started de novo.
The first attempt to avoid forming differences when interpolating
in a table not provided with them, and at the same time to escape
76 INTERPOLATION [3.7
the labour of Lagrange’s formula, was due to C. Jordan,* who
formed certain linear interpolates and operated upon these. We
shall not describe Jordan’s process, since an essential improvement
thereon has been made by A. C. Aitken,t who reahsed that the
practical advantage lay in the process of linear interpolation, and
devised a method of interpolation by iterating this process.
3-81. Aitken’s Linear Process of Interpolation by
Iteration. Let u^, u„ ..., denote the values of a function
corresponding to the arguments a, b, 0, ... .
We denote as usual the divided differences by [ah], [abc],
Let /(a;; a, 6, c), for example, denote the interpolation polynomial
which coincides in value with at the points a, h, c. Then by 1-9 (2)
a, h) = + (a; — a) [ah] ,
f{x] a, h, c) = Ua + (®-a)[ah] + (a:- a) (a; - h) [ahc],
/(a:; a, h, c, d) = Ua+{x-a)[ah'] + (x-a){x-b)\abc]
+ (a; - a) (a; - h) {x - c) [abed] ,
and so on. We have then, for example,
/(a:; a,b, c,d) =f[x; <^,b, c) + {x~a)(x — b){x-c)[abcd].
Since the order of the arguments is immaterial we have also
fix; a,h,c,d)=/(a;; a, h, d) + (a; - a) (a; - h) (a; - d) [ahed].
Eliminating [abed] we obtain
(1) fix ; a, h, c, d) = (x,b,c)~ic-x) fixj^, b, d)
id-x)-io-x)
^ fix; a, h, c) c-x I . , , ,
fix; a, h, d) d-x | ‘
^ Thus /(a; ; a, h, c, d) is obtained by the ordinary rule of propor-
tional parts from the values of fix ; a, h, y) for y = c, w = d. This
argument is clearly general.
(1) AUi del Congrmo Internaz. dei Matematici, JBologna, (1928), vi, p 157
(u) Metron, vii (1928), p. 47. wi.
t A. C. Aitken, Proc. JEdinhurgh Math JSoc. (2), iii (1932), p. 56.
mTERPOLATION
3-811
Applying this rule we can now write down
77
the following scheme :
Argument Function (1) (2)
a Ua
h Ms a,b)
c j{x ; a, c) f{x ; a, b, c)
d Ug, f{x ; a, d) f{x ; a, b, d)
(3) ... Parts
a-x
h-x
c-x
f{x, a, b,c,d) d-x
Each entry is formed by cross-multiplication and division, with
the numbers in their actual positions, thus
f{x; a, 6)=
I %
f{x\ a, c) = ““
a-x
b-x
(b-a),
a-x
c-x
~ (c-a),
a-x
d-x
{d-a),
f(x ; a, 6, c) ••
f{x ; a, b) b-x
f{x; a,c) c-x
(c-6),
and so on.
The above scheme constitutes Aitken’s process.
The members of column (1) are linear interpolation polynomials,
those of column (2) quadratic interpolation polynomials, those of
column (3) cubic interpolation polynomials, and so on. If a
numerical value be substituted for a;, each member of the rth column
is the value of an interpolation polynomial which coincides with u^.
at r-i- 1 points and gives the value of within a degree of approxi-
mation measured by the remainder term at this stage. The process
is therefore completely equivalent to interpolation with Newton’s
general divided difference formula. Thus, for example,
=/(cc; a, 6, c, d)+ {x-a){x-h){x-c)[x-d)uf^^ j 4:\ ,
where 5 lies in the smallest interval containing a, 6, c, d, x. If then
interpolation by Newton’s formula be practicable, the numbers in
later columns will tend to equality as the work proceeds. This leads
to a simplification, since in the linear interpolation those figures at
the beginning which are common to all the members of a column can
78
INTEBPOLATION
[3-81
be dropped. Tte process terminates when further interpolation
would cease to influence the result. With regard to the column
headed Parts/' we may replace the entries by any numbers pro-
portional to them, as is obvious from (1). In particular, if the argu-
ments be equidistant, we may divide each entry in this column by
the argument interval co. Moreover, when the arguments are equi-
distant, this division by ce will make them differ by integers. The
method is eminently suited to use with an arithmometer and is
independent of tables of interpolation coefficients. The process can
also be used at the beginning or end of a table.
Example. From the given values of the elliptic function sn (ir | 0*2),
find by interpolation the value of sn(0*3 | 0*2).
z
sn {z 1 0-2)
(1)
(2)
(3)
(4)
(5) Parts
0-0
0-00000
-3
•1
•09980
29940
- 2
•2
•19841
29761-5
29583
-1
•4
•38752
29064
29356
..469-5
+ 1
•5
•47595
28657
29248-5
..471-5 .
.. 467-5
+ 2
•6
•55912
27956
29146-4
..473-85 .
..467-3 .
..7*9 +3
Here the parts ” are - *3, - *2, - *1, -{- *1, -h *2, + *3, which we
replace by integers. We also treat the tabular numbers as integers
and carry extra figures as a guard. After column (2) we can drop
the figures 29. We could likewise treat the entries of column (3) as
9*5, 11*5, 13*86. The following are examples shewing how the
numbers are obtained.
29940 =
0
9980
-3
-2
-1,
29064 =
0
38752
-3
+ 1 •
469-5 =
583
356
-1
+ 1
^2,
7-9 =
7-5
7-3
■1.
The result is 0*29468, which is correct * to five places.
3*82. Aitken’s Quadratic Process. Suppose given an even
number of symmetrically placed data such as
* Milne-Tbomson, Die elUptischen Funktionen von Jacobi, Berlin (1931).
INTERPOLATION
79
The expression
[y+x)u,j + {y-x) u^y ^ . ^2^
is an even function of y, since it remains unaltered when -i/ is
written for y. This justifies the notation. Also/(x ; x^) = % . With
the given data we can form., by means of (1), the values off{x ; a^),
f{x\ b^),f{x; c^). If we apply the linear process of the last section
to these new data, taking as variables a^, b^, we thereby obtain an
interpolated value for/(cc; x^) or u^. Thus we can form by 3-81 (1)
(2) -(‘’-O').
f, 2 ;■> fix; a\b^) b^-x^
/(x;»> J-,o-)= ^ ^
and so on until the data are exhausted.
Thus we form from (1)
f{x; b^, c^)
{b^-a%
~ (c2-62),
f{x; a^)
(2a), etc.,
and form by means of (2) the scheme
f(x; a^)
fix ; ¥) f{x ; a2, b^)
fix; c2) fix; a^c2) /{x; a\b\c^)
which is essentially the same as that of the last section, but with
squared variables.
Since we are in fact using 2, 4, 6, ... values of the function in
successive columns, we are progressively taking account of the
first, third, fifth, ... differences in an ordinary interpolation formula.
If then 2n values be used, the remainder term is
B,,ix) = (x^-a^) ix^-b^) ...
This process can also be worked at the end or beginning of a table.
In applying this method to equidistant data we first subtract
from each argument the middle argument m about which they are
80
INTERPOLATION
[3-82
symmetrical. After division by the interval w the arguments
become ± ± ±2|, ... and x is replaced by the phase
p = {x-m) j ci>. The “parts” then become
-which are of the form 0, 2 + 0, 6 + 0, 12 + 0, 20 + 0,
30+6,..., where 0 = (|')®-p®. The advantage of this will be
apparent from the following example.
Example. Prom the following values of Jacobi’s Zcta function
Z{x \ 0'6), calculate by interpolation Z(0-11 | 0-6).
X
Z(a:|0-6)
(a)
(i)
0-00
0-0000 000
-2-5
2-75
-04
-0133 469
-1-5
1-75
•08
•0266 172
-0-5
0-75
■12
•0397 350
+ 0-5
o
1
•16
•0526 262
+ 1-5
-1-25
•20
•0652 186
+ 2-5
-2-25
Here the middle argument is OTO, column (a) shows the prepared
arguments obtained by subtracting OTO and dividing by « = 0-04.
The phase is (OTl - OTO) +- O-Oi = 0-25.
Column (6) gives the numbers x-a, x+a, etc. for formula (1).
Thus we have (treating the tabular values as integers),
364 655'50 OT875
362 598-25 3647 38-99 2-1875
358 702-30 3647 38-41 ...9-31 6-1875
The first column is formed by formula (1) thus, for example.
364 555-50 =
397 350
266 172
After this we use (2) thus.
--25
•75
-+1.
364 73841=®’“““-“ 6
358 702-30 6-1875 , '
Finally we obtain Z (0-11 1 0-6) = 0-0364 739, which is correct to
7 places.*
M Zeta Function of Jacobi, Proc. Roy. Soc., Edinburgh,
(iyo2), p. 2So.
INTERPOLATIOlSr
81
3-82]
If the number of data be odd but symmetrical, Aitken has devised
several methods founded on iteration, but it is actually simpler to
retain one method and annex an extra tabular value. For details
the reader is referred to Aitken’s paper.*
3-83. Neville’s Process of Iteration. A somewhat different
technique has been developed by Neville,*!* which has the advantage
of finding a place in the iteration scheme for those derivates of
the function of which the values may be known.
The essential point of the process consists in interpolation between
consecutive entries in the columns, beginning at the centre and
working outwards, new functional values being adjoined as required.
The clustering of the interpolates round a central value leads to
greater equality between the members of a column as the work
progresses and avoids the necessity of any preliminary estimate of
the number of tabular values required.
With the notation of 3-81 the process is indicated by the following
scheme :
Argu- Funo-
raent Parts tion
a X- a
fix ; a, h)
h
x-b
a, 6, c)
fix ; b, c)
fix ; a, b, c, d)
c
x-c
u.
/(*;
bjCjd) /{x; a,b,c,d,e)
fix ; c, d)
fix ; b, c, d, e)
d
x-d
fi^l
c, d, e)
fix ; d, e)
e
x-c
(1)
Here it is convenient to write 3*81 (1) in the form
x-a f(x ; b)
f{x ; a, h, c)
x-c f{x; b, c)
(c-a)
* he. cit. p. 76.
t E. H. Neville, in a paper read at the International Congress of Mathe-
maticians, Zurich, 1932. This paper will be published in a commemoration
volume of the Journal of the Indian Mathematical Society. Prof. Neville has
kindly allowed me to use his MS. and upon this the present section is based.
INTERPOLATION
82
[3*83
in order that the '' parts ” may be identified as lying at the base of a
triangle of which the interpolate is the vertex.
The process of course leads to the same interpolation polynomial
as Aitken’s process when founded on the same arguments.
In the case of equal intervals the parts may be most conveniently
treated by division with the tabular interval co.
Example. Find sin 0*25 from the values given below.
0-1
1-5
0-0998
2481-5
0-2
■1987
...73-6
2471 4-1
0-3
-•5
■2955
... 74-G
2485-5
04
-1-5
■3894
Here x =
0-25, (0 =
0-1; the
parts are given in tlie second column.
2471
•5
1987
.K 71
74-6 =
-•5
2955
-1-5 85-5
sin 0*25 = 0*2474.
In order to introduce derivates into the scheme, we first notice
that the interpolation polynomial /(a;; a, b) is given by
f{x; a, 6) = +
where [ah] is the divided difference of .
If a = 6, we have [aa] = Ua (see 1*8 (2) ) , and we can write
(2) f(x; a, a) Ua-b{x-a)Ua = f(x; a^) say.
Similarly, ita—b — c, we have
(3) f{x; a, a, a) = f{x ; a^) + (a; - af [aaa]
= f{x; a^) + (x-a)^ u'a = f{x ; a^) say.
These values can be calculated and introduced into Neville’s
scheme in the appropriate columns. Suppose, for example, that we
3-83]
INTEBPOLATION
83
are given u^,, u'^. The scheme becomes, by repeating
the arguments,
a x-a
f(x; B?)
a x-a f(x; a^)
f(x; a2) J{x-,a?,h)
a x~a Ua f{x;a\b) f{x;a^,b,c)
fix; a, b) f{x ; a% b, c) f(x ; a*, b, c^)
b x-b u, f{x;a,b,c) f{x;a\h,d^)
fi^->b,c) f{x;a,b,c^)
c x-c
f(x; c2)
C x-c Uf.
The entries in heavy type are calculated from (2) and (3) and the
interpolation proceeds by formula (1). Thus, for example,
fix; a?, b) =
x — a
x-b
fix; a^)
fix; a^, b)
~ib-a).
f{x ; a, b, c2) =
X-a
x~c
fix; a, b, c)
fix; b, c2)
d- (c-a).
Example. Find sin 0-25 from the following table :
X sin X cos X
0-2 0-1987 0-9801
0-3 -2955 0-9553
We form /(0-25 ; 0-2^) = 2477-05, /(0-25; O-S^) = 2477-35, then
0-2 -5 1987
2477-05
•2 -5 1987 ...4.0
2471
•3 --5 2965 ...4.2
2477-35
- -5 2955
sin 0-25=0-2474.
-3
84
INTERPOLATION
[KX', III
EXAMPLES III
1. Find approxiniatelj the value of antilog 0-9763 452 given
the table :
X
antilog X
0-95
8^9r2 509
•96
9^120 108
■97
9^3:i2 543
■98
9^549 926
■99
9^772 372
and discuss the limits of error. Calculate also antilog 0*9532 64 1 , and
antilog 0-9873 256 (a) by Newton's formula, (b) by Aitkon's process.
2. The logarithms in Tables of n decimal places dilfer from
the true values by i 5 x at most. Hence shew that the
errors of logarithms of ^places obtained from the Tabh‘-s by
interpolating to first and second differences cannot exceed
± 10“”+ c, and ± 10“” x (9/ 8) + e' respectively, e and e/ l)eing the
errors due exclusively to interpolation.
[Smith's Prize.]
3. From the table
X
Jx
530^1
23-02
3901
540-1
23-24
0052
550-1
23-45
4211
560-1
23-66
6432
form a scheme of differences and calculate 7^^0-67459, 7040*67459,
7550-67469, in each case determining the limits of error.
4. If f{a) = Uq, /((X + co) = tij, prove the following formula for
interpolation in the middle, or to halves ",
/(a + l-co) = [xwj - g ™ j
- + fat (^) ■
Obtain tbe general form of this result when the last difference used
is
KX. Ill I
INTRRrOLA'l'rONr
5. Supply the values correspoading to x = O-iOi, -103, -105 in
the following table :
X sin X
0*100 0*09983 3417
‘102 -10182 3224
•104 -10381 2624
•iOG *10580 1609
6. The following table gives values of the complete elliptic
integral * E corresponding to values of m ( = F) :
m
E
0*00
1-5707 96327
*02
1-5629 12645
*04
1-5549 68546
•06
1-5469 62456
•08
1-5388 92730
Insert the values corresponding to m = 0*01, -03, *05, *07 and
construct a corresponding table of E for k = 0*00, *01, *02, , *08.
7. Find expressions for the remainder term in Stirling’s formula
when terminated with a difference of odd order, and in Bessel’s
formula when terminated with a difference of even order.
8. Taking o) = 1, prove the central difference interpolation
formula t
/(^)=/(0)+ v^(
s i ^ ^
X /"x-h ln- 1
} X fx+ ^s ~ 1
^-1
s>/(0)
X fx+^n-^
9. Taking co = 1, prove the equivalence of the following opera-
tions :
i>.u, = l(E^ + E
* L. M. Milne-Thomson, Proc, London Math. Soc. (2), 33 (1932), p. 163.
t Steffenson, Interpolation^ p. 32.
86
INTEEPOLATION
iKX'. HI
10. Use tlie result of the last example to prove tliat the to'rms
of the central difference formula of example 8 are obtained by
expanding in ascending powers of S the exprovssiou
/l8+(l+iii=)*}'’/(0).
11. Find from the following data an approximate' value of
log 212 :
log 210=2-322 2193 log 213 = 2-328 37‘K)
log 211=2-324 2825 log 214 = 2-330 -Ib’nS
and discuss the error term.
12. From the following table
mately log F (|) :
of logr(w.),
d(‘t(‘nniiu’
n
logr(n)
n
log r(«.)
2
12
0-74556
iii
0-18432
y
1"2
-55938
M
1 ‘J.
-13165
i
1'2
-42796
\)
1 2
-08828
y‘-j
-32788
1 0
-0526 1
13. If n radii vectores (n being an odd inti-ger) be drawn from
the pole dividing the four right angles into equal parts, shew that
an approximate value of a radius vector «<,, which makes an angle 0
with the initial line is
_ 1 ,^sin|n(G-a)
n2j sin -1(0 -a)
where a, , are the angles which the n radii vectores make with
the initial line.
CHAPTER IV
NUMERICAL APPLICATIONS OF DIFFERENCES
In this chapter we consider a few important applications of
differences a-iid iifferpolation formulae, mostly of a numerical
nature.
4*0. Differences when the Interval is Subdivided.
Suppose that we ha,ve the difference scheme
§•« Mo
and that we wisli to form
the table u . j, u . o, ^l . jj,
subdivided into ]() equal
scheme will t>he.n read
tlie central differences corresponding to
, ti, 0, where the interval has been
])arts. The first two lines of the new
Uq
a- Mo
d^UQ
5«-05
05
M-l
U .jL
9“^ u .j
We have du .05 = tc,i - u
0, SO that from Bessel’s formula
(1) . 05 = i\> u <1 »)
\ 0 0 0 + KooU P-S
The remaining first differences can be found in the same way.
To form the second differences we have
87
88 NUMERICAL AURLICATIONS OE 1)1 FEKRENCES [-H)
Using Stirling’s formula., this gives
(2) rJoS^Mo- 4«¥..« S^«o+--
Similarly from
u -05
we obtain
u .Q5
and in a like manner
9^ Uq = i 0 0 "*^0 ”" * * * •
It will be noticed that division of the interval by 10 has the
general effect of reducing the order of magnitude of tlie first, second,
third, differences in the ratios 10“-^
More generally, if we subdivide* the interval into a ])arts,
we shall reduce the differences approximately in the ratios
4*1 . The Differences of a Numerical Table. The succ(‘ss
of interpolation by the formulae of Chapter III in a numerical table
of a function tabulated for equidistant values of the argument
depends upon the remainder term becoming insignificant to the order
of accuracy required. Since the remainder terra is proportional to a
value of the derivate of a certain order of the funcf.ion, and the
differences of this order are also proportional to values of this
derivate, the practical conclusion is that the effect of the differences
of a certain order shall become negligible. In the case of a poly-
nomial of degree n, the (n-f-l)th order differences are zero ; in the
case of other functions, or even in the case of a polynomial wlum the
values are curtailed to a fewer number of figures than the full value
for the arguments, the differences never attain the constant value
f a* table in which it is proposed to interpolate
by differences it is therefore first requisite to ascertain at what stage
= w .2 - 3^0 ~ ,j,
“ 1 0^0 0 ii 0 0 00 — ... ,
application of the differences of the subdivided interval to
ra'Cf oU Brituh
4-1] NUMERICAL APPLICATIONS OP DIFFERENCES 89
the effect of the differences become negligible, which can be done
by actually forming the differences in question. Consider, for
example, the following table of Jx.
X Jx
A A^
X Jx
A
A^ A^
1000 31-02 2777
1010 31-780497
-8
10807
15729
0
1001 31-G3 8584
-7
1011 31-79 6226
-8
15800 -2
15*721
-tl
1002 31-65 4384
-9
1012 31-81 1947
-7
15791 -f2
15714
-1
1003 31-67 0175
-7
1013 31-82 7661
-8
15784 - 1
15706
0
1004 31-68 5959
-8
1014 31-843367
-8
15776 0
15698
0
1005 31-70 1735
-8
1015 31-85 9065
-8
15768 0
15690
0
1006 31-71 7503
-8
1016 31-87 4755
-8
15760 + 1
15682
-fl
1007 31-73 3263
-7
1017 31-89 0437
-7
15753 -2
15675
-1
1008 31-74 9016
-9
1018 31-906112
-8
15744 -f2
15667
-fl
1009 31-764760
-7
1019 31-921779
-7
15737 - 1
15660
1010 31-780497
-8
1020 31-93 7439
Here we see that the differences A ® do not vary much, while
alternates in sign. Since the third order derivate of Jx has a
positive sign, the fluctuation in sign of A ^ ni’ist be attributed to the
fact that the values here given are only approximations to Jx.
This suggests that we should investigate the nature of the fluctua-
tions which will be introduced into the differences by an error in the
tabulated functioE.
90 KUMERICAL APPLICATIONS OF DiFFFHFNCFS f.M
The effect of a single error x in an otherwises correct i ahhs is shewn
in the following scheme :
Error
/j
.cC
.1^
0
X
X
0
X
X
- Gx
Gx
0
X
X
-Zx
- 4x*
r KU
j lar
X
-x
■~-2x
H~ 3ir
+ Gx
- lOu;
- 2{Kr
0
X
-X
— *1.7;
-i r>x
1 i5.r
0
X
- Gx
0 a:
It will be noticed that the coeiBficients of th(‘ (‘rrors in Hie eoluiuu
/I ” are the coefficients in binomial expansion of ( I ■ as is indetsl
obvious from 2*5 (2).
If we replace the zeros in the above sclunuc b}^ ;/’j, j\>, j\,, x^,
we have for the sixth order difference opposite, to x t he expression
~ iCj- 6^2 + 153^3-- 2();r -1- Ibrj
In a table of approximations correct to a giv(*n nninber of figures
the maximum error in a single tabular valiui is fO-b, the ta[)ular
values being regarded as integers. The corresponding rnaxiinuni
error in the sixth difference arises when the toTors a, re alt ernatidy
+ 0*5 and -0-5, the result then being Wdnui the
differences fluctuate in a way which cannot be aetamnied for by tliese
considerations the presumption is that the t;al, Hilar values contain
an error, the probable position of which is indicated approximab.dy
as that entry which stands on the horizontal line opjiosite to the
largest anomalous difference of even order.
Returning to the above table of Jx, we see tliat a knowledge of
all the values in the column /J of the first entry in column /j and
+•1] NUMRRTUAL APPLICA'I'IONS OF DIFFERENCES 91
the eight-iigure value of KKMl, would enable us to reconstruct
the table, by first (completing the column ^ and then the column Jx.
Moreover, a knowledge of the last digit in the column Jx enables us
to infer the valines in the column thus :
v'^ A
7
7
4 -7
0
•1 -9
1
5
provided tliat we subtract in the a-ppropriate directions and have
prior knowItHlge of the a])])roxiinat(‘ magnitude of the numbers
This ftict is oiten ustdul in d(‘t(‘rniining whether a printed table
contains errors otluu- than thos(‘ in the hist digit. For we can rapidly
form the difhu’cncfhs j “ (or a higher ord(n’,if necessary) by differencing
the end digits in tlu^ maniK^r (h'seribed. We can then build up the
table again, prtdhraiily on a,n adding machine, and compare the
result with the original tabh\
4*2. Subtabulation. T1h‘ principle enunciated at the end of
4-1 can b(i (»m]>loyml in subta,hulation. Suppose a function to have
been calculated at ccjual inbu'vals of the argument and let it be
required to re< luee tlu* int(M-val to 1/10 of the original interval. The
proljlem here is to obtain new values of the function for the phases
•], '2, dl, -4, *5, -6, 7, *8, -9.
Taking Besstd’s formula, in, t)m modified form if necessary (see
3*41), we have
ip + 1 ) P (p - 1) (p - 2) Mj,
where the phase p lias the above values and the remainder term is
neglected. The formula has been written with the mean differences
14*2
92 NUMERICAL APPLICATIONS OF DIFFERENCES
doubled in order to avoid divisions by 2, The values of cotdii-
cients of the differences are shewn in the following table :
T
ipb-1)
Uv+i)Hp-i)
•1
- -0225
+ ■006
+ •0039 1875
•2
- -0400
+ •008
+ •0072 0000
•3
-•0525
+ •007
+ ■0096 6875
•4
- -0600
+ ^004
+ •0112 0000
•5
- -0625
•000
+ ■0117 1875
■6
- -0600
-•004
+ •0112 0000
•7
- -0525
-•007
+ •0096 6875
•8
- -0400
-•008
+ •0072 0000
•9
- -0225
-•006
+ •0039 1875
As we only want the last digit of each interpolate wc need only
write down the two relevant figures of the products in (1), keeping
one decimal as a guard. Those products which are negative can be
made positive by the addition of 10*0. If wc add the resulting
products for each value of p, keeping only the last two figures in
the sum and then round off, we get the required end digits of the
interpolates. The differences can then be formed and the 9 inter-
polates built up by summation as described in 4*1, the initial first
difference being obtained from 4*0 (1).
We shall illustrate the method by constructing the t.able of ^
given in 4-1 from the following data :
X
A
A^
A ''
■Uq
1000
31-62 2777
-792
157720
-1570
4-14
%
1010
31-78 0497
-778
156942
- 1546
+ 10
“2
1020
31-93 7439
-768
From (1) we have for the first interpolate, regarded as an integer,
(2) 3162 277 7-0+1577 2-0 + 3 5-3 + 0-1
= 3163 858 4-4 = 3163 858 4.
4-2] NUMERICAL APPLICATIONS OP DIFFERENCES 93
Writing down only the figures in large type we have the following
scheme in which the figures just obtained are in the horizontal line
opposite the argument 1001.
EAA^
1000 a b c d s
7
1010 Cl s 7 -8
9
1 7-0 2-0 5-3 0-1 i-i
4
-7
1 7-0 4-2 4-8 0-1 6-1 6 -8
0
2 7-0 4 0 2-8 0-1 3-9
4
-9
2 7-0 8-4 1-8 0-1 7-3 7 -7
1
4
3 7-0 6-0 2-4 0-1 5-5
5
-7
3 7-0 2-6 1-2 0-1 0-9 1 -8
6
4 7-0 8-0 4-2 0-1 9'3
9
-8
4 7-0 6-8 2-8 0-0 6-6 7 -8
6
8
5 7-0 0-0 8-1 0-0 5-1
5
-8
5 7-0 1-0 6-6 0-0 4-6 5 -8
0
6 7-0 2-0 4-2 9-9 3-1
3
-8
6 7-0 5-2 2-8 0-0 5-0 5 -8
0
2
7 7-0 4-0 2-4 9-9 3-3
3
-7
7 7-0 9-4 1-2 9-9 7-5 7 -7
3
5
8 7-0 6-0 2-8 9-9 5-7
G
-9
8 7-0 3-6 1-8 9-9 2-3 2 -8
4
7
9 7-0 8-0 5-3 9-9 0-2
0
-7
9 7'0 7-8 4-8 9-9 9-5 9 -7
7
0
1010
7
-8
1020 9
In the above scheme the numbers in the columns a, b, c, d repre-
sent the contributions of to the last figure of
the interpolate. The numbers under s are the sums of these contri-
butions, two figures only, and the column E represents the rounded
value of s. The columns refer in the same way to the
initial value We then form the differences as shewn. To form
the leading first difference we have, from 4*0 (1),
3^.05 = 15807-4 = 15807.
We can therefore complete the required table in the manner
described in 4*1. The theoretical value of the second difference
NUMERICAL APPLICATIONS OF BlFFI^RENi'KS [4.2
opposite the argument 1010 is by 4*0 (2) equal to - 7-78 or 8, which
agrees with the value in the scheme and serves as a elu^ck. ft will
be observed that the above process, if correctly |)eri\)rme.(i, nuist
reproduce the exact value of
If we had continually to reproduce cahnilations of the typr^ (2)
above, little would be gained by this procedure. It, is, how(‘ver, a
simple matter to construct, once for all, tallies wliitdi give th(‘ two-
figure numbers used in this process, for all values of tlu^ di ITt‘nuic(‘s
which can arise. Such tables, with examples of their usi% are t o bo
found in the Nautical Almanac, 1931. We may renmrk that in
practice it is more convenient to arrange woi-k so that the
additions, here shewn horizontally for conveuiencti of (‘X|)osition,
are performed vertically.* The decimal points an% of cours«^, un-
necessary, as in similar work of this kind.
Another method of subtabulation which lias been widely used
consists in calculating by the formulae of 4*0 tlu‘ theon*tieal valiuxs
of the differences of the interpolates in that differeiu‘.e column wlu^re
the differences are small or constant. The practical object ion to tliis
method is that small errors in a high order diifi'nuH'e ra{>idly a<‘(,‘umU“
late large errors in the functional values, so that a large number
of useless figures have to be carried tlirough th(:‘ work and subse-
quently discarded. If, however, the contribution of tln^ 1 bird crdcu’
difference in the original table be negligible, it is quite pra(‘ti(‘al to
assume a constant value for the second ditT<u’eru‘v and rtqtaq tlui
decimal figures of the interpolates, treating tlu^ original valu(\s as
integers.
Thus in the example just considered for Jx, st-arf ing with
a; = 1000, we have with the notations of 1*0, n(‘glecting tliird
differences entirely,
(3) , , a.05 = 15807*325,
and if we take the constant value
(4) a'l = ri-o = 7'850,
we can build up by summation the values already obtaiiuni. The
* For an example of extensive interpolation in t.liia way, sec L. M. Milno-
ThoiMoi^ 8tandmd Table of Square Roots (1929). This table woa iirsl formed
as a ten-ngure table and was afterwards reduced to eight figures.
4*2] NITMEKICAL APPLICATIONS OP DIFFERENCES 95
value for ^ 1 010 will be reproduced exactly, since the value obtained
with the above differences is
Wq + 10 (-IS - -045 fxS- u^) + 45 x *01(rS = Uq+Su^ = u^.
We can then proceed to calculate and start again.
The work can, however, be made continuous by using a suitable
second difference opposite %. Consider the scheme
-9 9.1
9.05 + 9 95
'ih ^
^ro5
which shews the end of the first calculation, to and the
beginning of the next, to ^/o. If for x we put d\ we shall not in
general produce the correct value of as given by the above
method, for
9i.o5 - *1 - -045 = *1 - 4-5 dh,
d.Qr, *1 Stq ~ *045 yiS'^u^ = -1 - 4-5 9?^ .
But if w(^ put
X = 9i .05 — d.Qr^ — 9 9ri = i (95 + 9pi ) + ^\J { — 100 x -I (95 + di-i) } ,
we obtain the corr(‘ct value of 9i.o5 and the work can then proceed
with the second constant 9i.i until we reach when the second
difference opposite to u.y is again adjusted. The decimal figures
introduced in this way are discarded when the tabulation is
completed.
4*3. Inverse Interpolation. The problem of interpolation
briefly stated consists of finding, from a table of the function, the
value of the function which corresponds to a given argument. The
problem of inverse interpolation is that of finding from the same
table the argument corresponding to a given value of the function.
Thus if y be a function of the argument x, given the table
Argmuent Function
Vi
^2 ^2
96 NUMERICAL APPLICATIONS OF DiPFKHKNCKS [4.3
we require the argument x corresponding to a given functional
value y. A numerical table b)’' its nature determines a single- valued
function of the argument but the inverse function may very well be
many-valued.
ThuS; for example, a table of the function y h3 takes
the form
cc 0 1 2 3 4
y 3 0-1 0 3
and there are two arguments corresponding to y 0 (and in fact to
every value of y). This simple example shows that care is needed in
formulating a problem of the inverse ty])e which nuiy only Ixjcome
determinate when the range of variation of tlic argument is in some
way restricted.
A practical way of obtaining such n^striction is to form a rough
estimate of the required result and to confine tlu^ argumcmts of the
table to values in the neighbourhood of this estimate. Assuming
then that a determinate problem has been formulated, we proetHnl to
consider methods of obtaining the solution.
4'4. Inverse Interpolation by Divided Differences. The
given table, by interchanging the roles of the argunuuit x and the
function y, becomes
Argument
Function
2/1
[M2]
2/2
lUi'Mh]
yz
1.72.73 1
^3
where we have formed the divided differences
(?/, -■ 2^2), etc.
We then obtain
a: - *1 + (2/ - 2/i) [M2] + {y~ Vi) {y - y^ -f- ■ • • .
4*4] NUMEEICAL APPLICATIONS OP DIFFERENCES 97
where, if we stop at the divided difference the remainder
term is
iy-Vi) {y-Vi) — {y-y„) lyy-iy^ — y^-
This is a complete theoretical solution of the problem provided
that we have some means of evaluating the remainder term or, in
other words, of calculating the nth derivate of y with respect to x,
or an equivalent process. In practice this may present difficulties.
We can, however, estimate the suitability of the value of x by inter-
polating the original table and seeing how far the result agrees with
the given value of y.
Example. Calculate
dx
J.37 V (1 — x^) (^-f r
from the following table * of cn {u | |) .
cn(w|
u
0-44122
1-2
-1-34048
•36662
1-3
-1-32066
-•132
--18
•29090
1-4
-1-30685
-•091
-•16
•21438
1-5
-1-29836
•055
•13736
1-6
The required integral is the inverse function cn"’^(*37 1 1). The
divided differences regarding the left-hand column as the argument
are shewn. We have, therefore, the value
1-2 + *07122 X 1*34048 + *07122 x *00338 x *132
+ *07122 X *00338 x *07910 x *18 = 1*29550.
4*6. Inverse Interpolation by Iterated Linear Inter-
polation. The iterative methods described in the last chapter
‘ Mhne-Thomson, Die elUptUohen FunJetionen von Jacohi, Berlin (1931).
98
NUMERICAL APPLICATIONS OF I>IFFEHENCKkS
[4-r,
(3-81, 3*83) are very well adapted to iriv<‘ns(? inter|)olatioii when
several orders of differences have to taktni into account.. These
methods do not depend on the argiimtu.it proca'eding by e(|ual
steps, and hence we may interchange argument and fumdion in
the same way as before and so arrive at t!ie requirtui result by
the general (linear) iterative process.
IsTeville * has shewn that known derivates, at Ituist of tht‘ first
two orders, can be conveniently employed by means of tlie formulae
dx ^ I dy d^x d-y / /dy y^
dy'^ / dx' dy^ dx^ / W/:r/ ’
We give the following example as worked by Aitkcn. |
Example. Find the positive root of the equation
a;'7+28x^-480=:^0.
From a graph of y = 480 it is easily stuui that tlu^ root
is slightly beyond T9. We form the table given below and seek the
value of X corresponding to y = 0.
y
z
-25-7140261
1-90
-14-6254167
1-91
2 3189586
- 3-3074639
1-92
2952228
28 82864
+ 8-2439435
1-93
2716929
87312 84138
+20-0329830
1-94
2483678
91702 17 5
Since y = 0 the left-hand column contains the
the process. Thus
^*9^ -2571402(51
23189586 = .i. i K
1*91 - 146254107 ’
and so on. We obtain
oj = 1*922884153,
which is correct to ten figures.
us(Hi in
KtSSOOlM,
* loo. ciU p. 81.
due to W. B. DavfcH, Kducatwnal Ti»M,
(1924) 61 ' Wliittaker and Kobinson, Calculus of ObaervativKa,
4-6] NUMERICAL APPLICATIONS OP DIFFERENCES 99
4-6. Inverse Interpolation by Successive Approxima-
tion. This widely employed method proceeds as follows. By
linear interpolation a few figures of the argument are found, and the
values of the function for this and one or two adjacent arguments
are calculated. Using these functional values we find some more
figures of the arg\imcnt, and then repeat the process until it ceases
to yield figures different from those already obtained.
Exanifle. Find the value of m corresponding to j = 0-01 from
the following table,* which gives values of the nome 5 as a function
of the squared modulus lc“ = m.
m
A A^
A^
A^
0-12
0-00798 89058
71 40944
•13
•00870 30002
82195
72 23139
1887
•14
•00942 53141
84082
68
73 07221
1955
•15
•01015 60362
86037
67
73 93258
2022
•16
•01089 53620
88509
74 81317
•17
•01164 34937
As a first approximation
Using Gauss’ formula, we
find
m
-14787
•00999 96780
1
00
00
-01000 04112
The interval is now .1 / 1000 of the original interval, so that by
♦ L. M. Milne-TLomson, Journ, London Math* Soc*, 5, (1930), p. 148.
100 NUMERICAL APPLICATIONS OP DIFFERENCES [4-0
4-0 the second difference is negligible and we have, dividing 3220
by the new first difference 7332,
m = -14787 4392.
4‘7. Inverse Interpolation by Reversal of Series. The
relation between the function y and the argument x, wliich is
obtained from an interpolation formula by neglect of the remainder
term, can be written in the form
y-2/i = + + «3 ?'■■’+ +«»?’”.
where j) = (x-x^) j is the phase.
This (finite) power series can be reversed in the form
where *
h — ^ h — /) — ■“
Q. — ^ .
Thus we have
y-yi “2 (y - Vi? ^ iW - «i«3) iy ~ yi?
Taking for example BesseFs formula and neglecting fourth order
differences, we have
y-yi-p hi + \ if' - p) gS® yi + 1 (p® - 'ip^ -t- Ip) S® yi ,
and we therefore take
ai = (8-^-gS2+V.,S=>) 2/:.,
etj = (-|gS2 - .^S®) y. , = J,S» y, ,
and we then obtain p from (1).
The method is of limited application since the (^onvcu'geiu't* is often
slow.
* For the first 12 coefficients, see C. .E. van Orntrand, Phil, Mag,, May 1908.
A simple determinantal expression for the general coefficient in givt‘ri by M.
Ward, Rendiconii di Palermo, liv (1930), p. 42, fSee also G. J. Lidstone,
51 (1918), p, 43. » .
4-7]
NUMERICAL APPLICATIONS OF DIFFERENCES 101
Example. Find an approximate value of cotliO-6 from tke
following table.*
X
coth~^ X
A
A^
A^
1*85
0-6049 190
-40908
1*86
•6008 222
- 40352
"F 616
-16
1*87
•5967 870
OQ7 rco
+ 600
1*88
•5928 118
~ 6v 1
Taking = *6008 222, we have
2/ - yi = - 8222, - 40657, ag = 308, ag = - 2*7.
Substituting in (1), we get p = *20254.
Since co = *01 , we have therefore the approximation
coth0*6 = 1*862025.
EXAMPLES IV
If =/(^’-bI)“/(a:^), Zl/W =/(^’+ l0)-/(ic), shew
that
{i+Aiy^f(x) = (i+A)f{x),
and by means of this formula express the forward differences! off (x)
for unit intervals in terms of the forward differences for interval 10.
2. Obtain corresponding formulae connecting the differences
for intervals o) and wco, where m is a positive integer.
3. Obtain the central differences corresponding to one-fifth of
the tabular interval in terms of the central differences for the whole
interval
* L. M. Milne-Thomson, Atti del Cong, Iniernaz, d. Matematici, Bologna^
(1928), t. 2. p. 357.
fThis is essentially the problem of Briggs. See H. W. Turnbull, James
Gregory ”, Proo, Edinburgh Math, Soc,y (2) 3 (1933), p. 166.
102
NUMERICAL APPLICATIONS OF DIFFERENCES [kx. iv
4. Obtain the table of Jx in 4-1 from the vahu\s of Jx at
interval 10 by first halving the interval and tlum intcn'polatiiig to
fifths.
5. Taking logarithms
to seven figures
at interval 10 in the
neighbourliood of 350, find the logarithms
at unit intervals from
350 to 370.
6. Find cosech 3-63 from the table of inverst^ values :
X
cosecn—u;
. J“
0-052
3-6503341
3704
•053
3-6313121
3r)r»r)
•054
3-6126467
3135
7. From the following table of inverse s(‘eaiits ealeulate
sec 0-17856 :
X
sec“^a;
.p
1-015
0-1721329
1 962
1-016
0-1777050
1782
1-017
0-1830989
1 629
8. Calculate cosec 1-3957 from the following table of inverse
cosecants :
X
coaccr^x
.1^
1-016
1-3986634
19()3
1-016
1-3930913
1 TS1>
1-017
1-3876974
1 0)29
1-018
1-3824664
1497
9. Check the value of i
oo.sec~^ 1-016 ill (8)
from the table :
X
cosec X
1-393
1-0160 16(56
109
1-394
1-0158 3463
108
10. Check the value of t
3ec-i 1-016 in (7) by means of the table :
X
sec®
.p
0-177
1-0158 7162
108
0-178
1-0160 5387
108
EX. iv] NUMEKICAL APPLICATIONS OF DIFFERENCES
11. Calculate cot~^ 2-9883 from tlie table :
X
cot X
0-320
3-0175980
-322
2-9975074
-323
2-9875522
-326
2-9580402
12. Prove that if the linear iterative process of 3-81 (p. 76) be
applied to the divided differences
^’‘a-3
a-h ’ a-c ’ a-d ’ ’
the multipliers being b,c,d,..., the sequences obtained tend to the
derivate u' (a). Investigate the remainder after n steps.
[Aitken.]
13. Prove that if the quadratic process of 3-82 (p. 78) be applied
to the central divided differences
2a. ' 20 ’ 'Ac
the multipliers being a^ 6“, ..., the sequences tend to w'(0), and
that if Wj, be a polynomial of degree 2n+2 the value obtained in
n steps is exact. [Aitken.]
14. Prove that if the multipliers used in Example 13 be
6^-0)^ c^-co“, the sequences tend to the subtabulated central
divided difference {«„ - u_^) / 2o.
15. By means of the methods of Examples 12 and 13 above com-
pute the derivates at a: = 0-00, 0-10, of the function Z{x 1 0-6) from
the tabular values given on p. 80. [Aitken.]
CHAPTER V
RECIPROCAL DIFFERENCES
5*0. The interpolation methods hitherto considered are founded
on the approximate representation of the function to be inttupolated
by a polynomial and the use of divided dilfercnces or the ecjuivahint
formula of Lagrange. Reciprocal differences, introduced by Thiele,*
lead to the approximate representation of a function by a rational
function and consequently to a more general method of interpolation.
In this chapter we shall consider a few of the most important pro-
perties of Thiele^s reciprocal differences.
5'1. Definition of Reciprocal Differences. Let the values
of a function f(x) be given for the values Xq, ... , of the
argument x. We shall for the present suppose that no two of
these arguments are equal. The reciprocal difference of f{x), of
arguments Xq, cCj, is defined by f
(1)
pK^i)
which is the reciprocal of the divided difference The re-
ciprocal difference of three arguments thrflned by
(2)
P2 (^0^1^2)
. ... ''o +/(:, ).
p(xoa:i)-p(cr^a:.)
*T. N. Thiele, Inter'polationsrechmmg, Leipzig, 1900. 8ee also N. E. Nor-
land, Differenzeyirechnung, Berlin, 1924.
fThe order of the arguments within the brtujkots is immatt^rial, for it will
be shewn in 54 that reciprocal differences, like divided diih’imwm, are
symmetrical in all their arguments.
104
KECIPROCAL DIFFERENCES
105
5*1]
We have here denoted the order by a suffix, since P2(^o%^2) is not
formed by a repetition of the operation denoted by p. The operator
p does not obey the index law, neither is the operator distributive,
that is to say, the reciprocal difference off{x)-^g(x) is not equal to
the sum of the reciprocal differences otf(x) and g{x).
Proceeding to reciprocal differences of four arguments we
define
(3)
P3(^0^1^2^3)
P2 (^o%^2) P2 (%^2^3)
+ 9(x^x^)
and generally when we have defined reciprocal differences of n argu-
ments we define reciprocal differences of n+1 arguments by the
relation
(4)
Pn-l (^0^1 • * • ^n-l) ■“ P«-i {^1^2 * * • ^n)
+ pn-2(^1^2 *•'
Comparing this with (1), we see that
(5) Pn (^0^1^2 * • * ^rt) ~ 9 Pn~l (^0^1 * * * ^n-l) + Pn-2 {^1^2 * * ’
Reciprocal differences may be exhibited in a difference scheme as
follows :
^0
/(S^o)
p(2?0^l)
/(a^i)
p (XjX^)
P2 (^0^1^2)
P3 (XQX2X2X2)
P2i^^2^z)
P (^2%)
P3 (^1^2^3^4)
/(a%)
P2 (^2^3^4)
•
RECIPBOCAL DIFFERENCES
106
[5-1
As aa example, the following table shews reciprocal differences
of 1/(1 + a;^) :
X
i
1 + x^
0 1
1 i
2 i
3 tV
4 jV
5
P
-2
-10
4 4 2
0'
P2 P3 p4
-1
0
— 1
lor
40
140
— 1
Iff-
0
0
This table exemplifies the fact that the reciprocal (lifferences of a
certain order of any rational function are constant. In this case the
differences of the fourth order have the constant value zero.
5*2. Thiele’s Interpolation Formula. If in the formulae
of the last section we write x for Xq, we have successively,
^{xx^ = ^{XiX^) +
X — X^
^i[xXiX^) -f(x])'
(?^{xXiX^ = I?^{x^x^^ +
x-x^
g^{xx^x^^)-
x-x^
P^{XX^X^^X^) - ^z{:XyX.^X.i) ’
^^{XXiX^X^i) = +
x-x.
P5(^^l*2*3*4®5) P3(^1^2*3^4)
TiiTis we have fox f{x) the continued fraction
RECIPROCAL DIFPE: INCES
07
<Nr
* O. Perron, Die, LeJire von den Kettenbriichen, Leipzig, 1929, § 42.
108
RECIPROCAL DIFFERENCES
[5‘2
we obtain a rational function, expressed in the form of a partial
fraction, which agrees in value with f{x) at the points
a?!, x^, x^, Xq.
It- folio TO that Thiele’s formula gives us a method of obtaining a
rational function which agrees in value with a given function at any
finite number of prescribed points.
Example. Determine tan 1-5685 from the following table * :
X
tan X
Pi
P2
1-566
208-49128
0-000018208313
1-567
263-41125
10615733
- 0-00382
1-568
357-61106
05023108
•00276
1-569
556-69098
01430462
•00178
1-570
1255-76559
The required value is
357-61106+
-000005023108 +
=357-61106 +
0005
-0005
-357-61106- -00178
■0005
=435-47730.
000006421268
According to Hayashi’s table the last figure should bo 2.
The principal part of tan x near Jw being
1
iu-x’
Thiele s formula is suitable for interpolation, while the ordinary
difference formulae are not.
5-3. The Matrix Notation for Continued Fractions. A
convenient notation for defining continued fractions of any
number of dimensions and for developing their properties has been
* K. Hayashi, Sieben u. mdirsUHMge Tafdn, Berlin (1926).
RECIPROCAL DIFFERENCES
5-3]
109
described by Milne-Tbomson,* and is well adapted for ordinary
two dimensional continued fractions of tbe form
(1)
^2 J^3 ^4
It depends upon the rule for matrix multiplication, f namely,
LVi + 2/2^2 + 2/2^2 J’
which is essentially the row by column rule for multiplying
determinants. We also recall that equality of two matrices
implies equality of their corresponding elements. Thus if
Ic dj p d’
then f, b = q, c = r, d = s.
If Pn/?n denote the ?ith convergent of (1), we have the known
recurrence relations
(3) Pn = dnPn-l + bnPn-i,
~ ^n— 1 “t ^71 5'n— 2 >
and hence from (2)
YPn p«-l Pn-2ira„ 11
\-<l» ffn-lJ L?n-l qn-A\bn 0 J '
If we write Po = 1 , = we have by repetition of the - above
operation
\Pn ?n-l1 pi np2 llp3 H ... P«-l dP” H.
L?n ^ J L^2 ^JL^3 L^n-1 0 J L^n
Thus we are led to define a continued fraction as the continued
matrix product
[? i]C: J]K J][:: a ■
and this definition leads at once to the recurrence relations (3) and
is fully equivalent to (1).
* L. M. Milne-Thomson, in a paper at the International Congress of Mathe-
maticians, Ziirich, 1932. Proc. Edinburgh Math. Soc. (2), 3 (1933), p. 189.
t Turnbull and Aitken, Theory of Canonical Matrices (1932), p. 3.
RECIPBOCAL DIFFERENCES
no
[5-3
In particular the components p„, qn of f^e rath convergent are
given by
(5)
n r®”-! nr®”!
L?J Li oJU oj -U., 0.1 UJ'
5-4. Reciprocal Differences expressed by Determinants,
If we write for brevity
(1) y= /(aj). Va = /(*«)> Pa == P» (% 2^2 .. . x„, i),
the components of the nth. convergent of Thiele/s continued fraction
are given, in the notation of the last section, by
\pnmryi nrpi nrp^-yi nfpa-Pi n
UnWJ Ll oJLa:-a:i oJLa;-a;2 0jL*-a;3 oj '
^ r Pn-l “ Pn-3 f ~j rpn ~ Pn-a"!
oJLa:-a:„ J'
Consideration of this product shews at once that
?2n+l(^»)> ?2n+l(a:)> P2« W
are polynomials in x of degree n while {x) is a polynomial of degree
n-1, and that these polynomials are of the following forms :
(2) 3’2b(®) = + +
(3) ?2n(^) = ^0 + ^1^!+ •■• + &n-2a:"”^ + a:’*~'^ P2„. J,
(4) ?an+i (a:) = Co + Cl a; + C2 a:H . . . + c„_^ 3:”-H x” p,,. .
(5) ?2n+l (x) = do + dj^X + d20l^+...+ d„_i + x".
If we take the wth convergent of Thiele’s continued fraction as
an approximation to we have
■ (^)
' in (a;)
+iJ„ {x),
where (x) is the error of the approximation.
Now Rn (a:) vanishes when x = x^, x^, , x„, m that
Vi
in [X,) ’
s = 1, 2, ... , n,
(6)
and hence
W Tn {^a)-yain (X,) = 0, S = 1, 2. ... , n.
KECIPROCAL DIFFERENCES
111
54]
Thus from (2) and (3), writing 2';^ for n in (7), we have
(8) do + -f . . . + -b^y,- 6^ ...
- &«-2 ys - p2n-l = 0.
If in this relation we give 5 its values in turn, namely 1, 2, 3, ... , 2n,
we have a set of 2n linear equations, which sufi&ce to determine
aQy ^1, ... , 6„„2 p2n~i subsequently the value
T^n (^) / ?2n (^)- The chief interest, however, lies in the determina-
tion of p2n-iJ which we obtain by direct solution as the quotient of
two determinants. Rearranging (8), we have the equations
+ % ^s-W^'sys + <^2 ^s^-b^
+ x,«-i + X/ ^ y, = 0,
from which we obtain
(9) p2n-~lM2*--^2n)
” i ril/s, ^sVs^ ••• ) [
where the determinants are contracted by writing only the 5th row
in each, 5 = 1, 2, 3, ... , 2n. These determinants differ only the
last column.
The above expression gives the important result that
P2n--l(^p^2> - . ^2n)
is a symmetric function of the arguments : for an interchange of any
two arguments merely interchanges two corresponding rows in the
determinants and leaves the value of their ratio unaltered.
To obtain the value of we have similarly from equations (4),
(5), (7),
^0 - ^0 + Cl x,-d^x,y,+ .,.+ Cn_i
- dn^l ys - Vs + P2n = 9,
which gives in the same way
(10) P2n(^1^2-^2«+l)
1 1? Vs) ^S) ^sVsJ •'* J
> n—X rf w— 1
n—l
. n-l
whence we infer in the same way that P2n(^i% **• ^2n+i) ^
symmetric function of the arguments.
112
KECIPKOCAL DIFFERENCES
[5-4
Thus we have proved that the reciprocal differences of any order
are symmetric functions of their arguments.
It follows from this result that the arguments can be taken in any
order which may happen to be convenient. In particular, we could
write down interpolation continued fractions in which the reciprocal
differences proceed across the difference scheme along a zig-zag line
in complete analogy with the backward and forward formulae of
Gauss. We shall not develop this here, but it is worthy of mention
from the standpoint of practical interpolation.
5’5. The Reciprocal Differences of a Quotient. The
determinantal forms for p2„_i, p2„ furnish a means of obtaining ex-
pressions for reciprocal differences of a function which is expressed
in the form of the quotient of one function divided by another, say
f(x) I g (a^). If for brevity we write
y=fix), z=g{x), y,=f{xff z,=--g{x,),
we have from 5-4 (9), after multiplying top and bottom by
hh -
(1)
\ i/si ^3^31 ^sys! •
xf-H„ x,''~^y^,x--^z.,x;‘~\
y>
Similarly from 5-4 (10) we obtain
=■ \?.v ^>hi ^3y„ — , xfy, |
I ^5) Xs^3> ^3ys3 a:,"z, I '
Thus we have the following particular relations for n = 1, 2 :
.
1 \ Vi
^2 ^2 ^
^2
P2
H
Vx
^Vx
h
Vx
XjtZi
h
Vi
”T“
h
y%
XgZ^
^3
Vz
H
yz
X3Z3
5-5] BECIPROCAL DIFFERENCES Hg
% Vi ^1% % yi Xjij/i
P3 2:2 y^ x^y^
H Vz ^3% ^32:3 2:3 ^3 cCgSJg x^y^ ’
^4 2/4 ^42^4 ^4, y^ x^z^
H Vi %% ^i2/i ^2/i z^ y-^ x-^z^ ^\y\ ^1%
.y. H 2/2 ^2^2 ^^2 2/2 ^22^2 2!2 y^ x^z^ x^y^
94[^)— % 2/3 x^y^ xly^^ ~ 2^3 ^3 x^z^ x^y^ x^z^
h 2/4 ^4^4 ^42/4 ^42/4 2:4 2/4 ^42^4 ^42/4 ^4^4
2^5 2/5 ^5^5 ^6 2/6 ^52/5 2:5 2/5 ^5% %y5 ^^52=5
We can use these relations to prove that at a certain order the
reciprocal differences of a rational function are constant. .
To illustrate the reasoning, we take the function
a-\-bx -}- Gx^ _^y
a+ Pic+yx^ z
and shew that the fourth order differences are constant.
The determinant in the numerator of P4 is
N^= I a 4- 4- ya;/, a 4- hx^ 4- cxj^, ax^ 4- px/ 4- yx^^,
aXs+ 4- ccc/, aa:/ + bx^^ 4- ccc/ 1 .
If we denote the columns of a determinant by c^, Cg, C3, C4, Cg and
the columns of the new determinant, derived by manipulation, by
Cl , etc., we can form successively the following determinants each
equivalent to :
I 1^1^. + a 4- 6a;, + cxj^, ocx, + pa;/ + ya;/,
ax^ 4- 6a;/ 4- ca;,^, axj^ 4- bx^^ + cx/ 1
by the operation c{ = - - Cg ;
a
1 + Ti^/> ® + T®/. V/ + ca:/. + hx^ + ex* \
by the operation = Cj - 1- c,, “a - Ci> «/ = ^ c, ;
I PiOJ.+ Yia:/, a + c^x,^, p2a:/ + Y®A hiX,^+cx*\
by the operation 04=04-^^03, 05' = 05-^03;
a+<hP>\ ca;/|
114
BECIPROCAL DIFFERENCES
[5-5
Y /
by tie operation =
c.- “ r. ;
by the operation c^' = Cj -
so that
*^2 ■“ fj ‘ a ?
H2
N^= pi^PgCoC I .T^., ], a:/, X^K J'/I-
Similarly for the denominator we obtain
1, xj^, :^VM-
c
Thus p4
T
' , which is constant.
We have assumed in the above construction that none of
^3 Pi? ^2’ ^2’ T
vanish. These cases present no special difficulty, but we may note
that if y = 0, is infinite, so that pg must be constant.
6*6. Some Properties of Reciprocal Differences. If in
5-5 (2) we put t/ = 1, we have
(1)
I ^83 i? ^S3 ... , ^2,,, I
I ^S3 f J ^S^S3 ^S3 •** ? ^3^
1
P2n (^)
by 54 (10). Thus the reciprocal differences of even order of a given
function are equal to the reciprocals of the reciprocal differences of
the same order of the reciprocal of the function.
Again, from 54 (10),
P (v+ c) = I ys + o^s3 ’C, x/%, -f-
1 1. Vs^o, X,, X, ... h;, a:/*
= I ys3 ^33 ^S^~^y33 + I
ys3 ^33 I
Thus
P2n(2^ + c)= p2n(y) + <^*
Similarly, from 6-4 (9),
P2n-l(.y + c) = P2„_i(j/).
6-6]
BECIPROCAL DIFFERENCES
115
Again, from 5-4 (10),
/ A _ I 1. cx,y„..., cx»-^y„ cx,^y,
P2„v ^ cx”-^y„ 05," 1
(4) =cp,„(2/),
since the numerator contains the factor c in one more column than
the denominator.
Similarly, from 5-4 (9),
(5) p2n-l(C2/) = - p2„-l(y)-
Also, since
we have
a-{-hy (a-bc / d)
c + dy d c-^dy ’
+ /a-bcjd
c + dy) ~ c + dy *
from (2),
== 3+U-
j,* from (4),
, from (1),
d'\'^ dJ?,^{c + dyY
6 6c\ 1
'■■d'^^°'-d)c + dp^„{yy
from (2) and (4), so that
IK\ „ - "• + ^P2n(y)
^ ^®'‘Vc + %/ c + cip2„(2/)'
This formula expresses the differences of even order, of a linear
fraction of y, in terms of the differences of even order of y itself.
If we take advantage of the symmetry in the arguments of the
reciprocal differences we can also form the differences of odd
order in Thiele’s continued fraction by means of 5T (5).
Thus, for example,
P5 “ P3 (^1^2^3^4) ^ P5 (%^1^2^3®4^6) ““ 9z {^1^2^3^4)
= p P4(%^^1^2^3^4)j
so that from a knowledge of the even order reciprocal differences of y
we can expand
a+by
c+dy
in a continued fraction.
116 EECIPROCAL DIFFERENCKS [5.7
5*7. The Remainder in Thiele’s Formula. If we take n
interpolation points x^, x^, ... , x„ and form Thiele’s continued
fraction for a function f(x), we can write
(1)
fi^) =
Pnjx)
where p„{x), q„{x) are the components of the nth convergent.
R„{x) then measures the error committed if we replace /(a;) by the
«th convergent. Let (0,6) be the smallest interval containing
the real numbers x, Xg, ... , x„. Let us suppose that in the
interval {a, b) of the real variable x the function f(x) ha.s poles
at aj, a^, ... , «, of orders r^, r^, , r,, where ry\-u+ ... - m.
We shall suppose that none of these pole.s coincides with an
interpolation point and that at all points of (a, b) e.xcept the
poles f{x) has a finite derivate of order n. If we. write
(2) <l>{x) = {x-x^Y^{x-a.iY^ ... (x-a,)’’”,
the function f{x)<j>{x) is finite at every point of (a, h).
We shall suppose n to be so large that the degree of q„ {x) i.s greater
than or equal to m.
Now let a polynomial ({/(x) be chosen such that, if
(3) Q{x) = <j>{x)^(x),
Q{x} and q„{x) have the same degree. Thxis from 5-1,
if w = 2h, Q(x) is of degree /t - 1,
if n=2k+l, Q(x) 13 of degree k.
' Write
(4) E„(x) = i{x)
and consider the function
(x-Xi) (x-xY) ... {x~x„)
ynW Q(x)
f(f\ Pn{^) _ / \ ... {t — X„)
which vanishes when t = Xj, , x„ and abso when t = x by
(4) and (1).
Then the function
(5) £0(t) =f{t)qn{i)Q{t)-Pn{t)Q{i)-l{!!0){t-Xj) ...{t~-X„)
5-7] RECIPROCAL DIFFERENCES 117
also vanishes when I = x, x^, x^,..., x„, all of which lie in the interval
(a, 6). Thus by Rolle’s Theorem w' (t) has at least n zeros in {a, h),
co" (0 has (n-l) zeros in (a, b), and so on, until finally we conclude
that (t) has at least one zero, say f = ^, in (a, b). 'Sow p„{t)
is a polynomial of degree h, when w = 2A or 2A + 1, and we have
chosen Q{t) so that the degree is A - 1 or A, according as n = 2A
or 2A+1. Thus p„{t)Q(t) is a polynomial of degree n-1 and
hence the nth. derivate vanishes identically. Hence from (5),
whence we have the error term
(6) R„(x)
(x-Xj)(x
. ^ -
e(5)).
nl qn(x)Q{x)
It should be observed that the above formula is only valid if n be
sufficiently large.
If / (a;) have no infinities in the interval (a, b) we can take
Q(x) = q,,{x),
and the error term is then given by
(7)
{X) =
{x-x^) {x~x.^) ... {x-x^)
5*8. Reciprocal Derivates ; the Confluent Case. In the
definition of reciprocal differences we supposed the arguments to
be distinct. Just as in the corresponding case of divided differ-
ences, we can here suppose two or more arguments to coincide and
so obtain confluent reciprocal differences. The simplest way to
proceed is to consider the limiting forms assumed by the deter-
minants (9) and (10) of 54. Thus, for example, we define
P2 {xxy) = lim p2 {x, x+h,y)
1 f{x) xf{x) 1 /(a;) X
= lim 1 f(x + h) {x+h)f{x-\-h) 1 f{oo + h) x-^h
h—^o
1 f(y) yf(y) ^ fiy) y
BECIPROCAL DIFFERENCEkS
118
[5-8
Subtracting the first row from the second and dividing by h, we
have
1 /(^) ^/(^) i /(^)
P2(a;xy)= 0 f'{x) xJ'{x)^J{x) 0 /(a:) I
1 fiy) yf{y) i f{y) y
If now we write for y, subtract the sum of the first row and
h times the second row from the third row, divide by it- and then
let ic -> 0, we obtain
1 f{x) xf{x) 1 f{x) X
(1) P2(a;a;a;)= 0 f{x) xf{x)+f{x) 0 /'(:r} 1
0 0 irix) 0
It is clear that in this way we can obtain confine nt reciprocal
differences of any order, since indeterminate forms can always be
avoided by taking advantage of the syimnet.ry of the dilT(‘ren(‘es
with respect to their arguments. Particailar interest attaxdies to
the case in which all the arguments have a conunon value. Ilua
particular form of confluent reciprocal difference is called a
reciprocal derivate and we write
2) r„f{x)= lim p„{x^x.^...x„^.^)
= x).
In particular,
(3)
rf (a;) = lim —
I
*1, 3=2, -►ic/ (^l) ■”/ f' (^') ^
SO that the reciprocal derivate of the first order is the reciprocal of
the ordinary derivate.
The successive reciprocal derivates can be calculated from a re-
currence relation which may be obtained as follows :
from 5*1 (4), we have
Pg {xxx) - p2 (xxy) 1 ^
x~y '
p2 i^^y) - p2 jxyy) ^ 1
x-y Pz &mf- 9 i^yV
p2 im)-92 (yyy) ^
^-y 9z{^yyW-9{yyy
RECIPROCAL DIFFERENCES
119
5-8]
adding, we have
Pa {xxx) - Pa (yyy) ^ 1 ^
x-y Pa (xxxy) - p {xx) pa [xxyy] - p {xy)
+ __J: .
Pa i^yyy) ~ p (yy)
If we now let y we obtain
Thus
1 __ 3
‘Tzfix) = rf{x) + 3rr^f{x).
This is a particular case of the general recurrence relation whose
form is easily seen to be
(4) / {x) = rn-2 / (^) -t ^n-i f (^) .
In particular,
■7>)
which agrees with (1).
Since the reciprocal differences of some order of a rational function
are constant, the same must hold for reciprocal derivates.
For example,
rx^ = 2^^ , 7*2 x'^ = - 3a?^, x^ = 0.
fax H- 6 \
{cx-vd)^
fax-vh'^ _ a
Vco; dJ
be -ad'
Vex + d/ ~ c
This last result can be obtained also from 5-6 (6) as follows :
/ax-^b \ ___ ar^x+b _ a
\ccc + d/ cr^x + d^
smce CO ^
6*9. Thiele’s Theorem, We have seen that Taylor’s Theorem,
which gives the expansion of a function in a power series whose
coefiSicients are proportional to the successive derivates at a point,
can be obtained from Newton’s general interpolation formula with
120
RECIPROCAL DIFFERENCES [5.9
divided differences. In a similar way Thiele’s interpolation formula
gives rise to a remarkable development of a function as a con-
tinued fraction in which the reciprocal derivates at a point are
employed.
In fact, when a:^, x^, ... , -9- x, we have
lim { p„ {xjx^ . . . x„^j) - p„_2 (xjx^ . . . x„_j) } z= r„/(x)- r„_^ f (x)
by 5-8 (4). Thus Thiele’s interpolation formula of 5-2 yields Thiele’s
Theorem, namely,
(1) f(x+h) =f(x)+ *
rf(x)+
in which, if we stop at the nth partial divisor, this will be
The enor term is given by 5-7 (6), where ? lies in the interval
{x, x+h).
Just as Taylor’s aeries terminates when the function is a polv-
nomial, so Thiele’s Continued Fraction terminates when the function
IS rational.
Thus, for example,
2x
+ -
L
1
2x^
n (3; + A) + 6 -f 6 ^ Ji
e{x+h) + d ~ cxTd'^ (cx+d)'^ ^ \
ad — be ^ -hc
c{cx+dy
ax+b _ 6
cx+d~d'^ d? J
ad^bc J^~bcj JIU) *
5-9] RECIPEOCAL DIPPEEENCES 121
Example. By means of Thiele’s Theorem, find a continued frac-
tion for e®.
We have
re* = e""*, rg e® = ~ e*,
Tg e* = — 2e“*, e® = e®,
fj e* = 3e-*, »‘6 e* = - e*.
These suggest the results
’•2ne*= (-l)"e=', r2«+i«*= (-l)"(»+l)e-*.
Assuming these for n, we have by 5-8 (4),
+ + = (-l)’'+i(« + 2)e-*,
so that the results are established by induction.
Then
(2n + 1 ) r r2„ e* = r2„+i e* - r2„_i e® = ( - 1)» (2n + 1) e-»,
(2n + 2) r e® = e® - r2„ e® = 2 ( - l)"+i e*.
In Thiele’s Theorem, writing 0 for x, and x for h, we have
e® = 1 + -
1 + -
-2 + -
-3+-
2+-
5+-
■2 + ...-
We can write this so that the integers all have positive signs, and
we then obtain *
* _ 1 a: a: aj a; x x
e*_ i + _ _ _ ^
* 0. Perron, loc. dt. p. 107 (353, (20)).
122
EECIPROCAL DIFFERENCES
EXAMPLES V
[KX. V
1. Form tie reciprocal differences of a; x f x
2. Form reciprocal differences for the following table :
a; 0*010 *011 *012 *013 *014
cotha; 100*00 90*91 83*34 76*93 71*43
and calculate coth 0*01257.
3. Form reciprocal differences of a® and hence develop a* in a
continued fraction.
4. Prove that p2> p4 can be expressed in terms of divided differ-
ences as follows :
I \ 1
P2(^1^2^3) ■“
1(^2) [^1
[x^Xq] [Xi
P4(*A®S2=4%)
1/(^3)
[^^22:3]
[^3^4]
1 \
[x^x^x^x^]
[x^x^^^x^]
Obtain corresponding expressions for p^, pg, pg.
5. Obtain determinants for (a;), (x), (x), (x), the com-
ponents of the third and fourth convergents of Thiele’s interpolation
formula.
6. Obtain Thiele’s interpolation formula for five arguments
in forms which utihse reciprocal differences in the same relative
positions in the difference scheme as those employed in the forward
and backward formulae of Gauss.
7, Prove that
pp2„ i^-; = -(p2„yrpP2„y.
P P2w+1
©=
-pP2«4-iy
(P2ny)(p2„+2yy
8. Given tke reciprocal differences of y for the argunaenta
^3’ > develop ^ in a continued fraction as far as reciprocal
y
differences of order 6.
EX. V]
RECIPROCAL DIFFERENCES
123
9. If we denote by y', by y”, and so on, prove that
dx
'TzV
y
3f
y^
2!
2!
y'l
3!
t^y = y
y
y'
y”
21
y"
y”'
y'
y"
y”'
'2T
3!
2T
3!
y'"
yU
y"
y'"
t''
3!
4!
2!
3!
4!
and obtain analogous expressions for rj y, y.
10. Determine the order of constant reciprocal differences in the
case of the rational function where fm{^),fn{^) are
polynomials of degrees m and n respectively.
11. Shew that in the case of a rational function Thiele’s inter-
polation formula terminates and yields a continued fraction which is
identically equal to the given function.
CHAPTER VI
THE POLYNOMIALS OF BERNOULLI AND EULER
In this chapter we develop some properties of two classes of poly-
nomials, which play an important part in the finite calculus, namely
the polynomials of Bernoulli and the polynomials of Euler. These
have been the object of much research and have been generalised in a
very elegant manner by Norland.*
We shall here approach these polynomials by a symbolic method
described by Milne-Thomson f by which they arise as generalisa-
tions of the simplest polynomials, namely, the powers of x. The
method is applicable to whole classes of polynomials, including
those of Hermite. Considerations of space must limit us to the
discussion of only a few of the most interesting relations to which
these polynomials give rise.
6*0. The cp Polynomials. We define <l> polynomials
of degree v and order n by the relation J
(1) =
V -0
where /„(() and g(t) are such that for a certain range of x the
expansion on the right exists as a uniformly convergent series in t.
Putting a: = 0, we have
(2)
where is called a <j> number of order n.
* N. E. Ndrlnnd, Acta MaA., 43 (1920), pp. 121-196.
t L. M. Milne-Thomson, Proc. Lemdon Math. Soc., (2), 35 (1933).
t Observe that the notation does not here denote tho nth dorivate
of
124
6-0] THE POLYNOMIALS OP BERNOULLI AND EULER 125
If in (1) we write x+yiov x, we obtain
Equating the coefEcients of f, we have
= + a: (i) (y) + x^ (g) i>^%{y) + ... +x>' (^)
Putting y = 0, we obtain
which shews, ur less that ^^:\x) is actually of degree v.
Thus we have the symbohc equality
(3) +
where, after expansion, each index of is to be replaced by the
corresponding suffix.
The polynomials are thus completely characterised by (3), and
by the numbers defined by (2).
From (3), we have
(*) E #;*’(«) = »»?'+»)- = V
(6) =
•^a V+ 1
Thus differentiation depresses the degree by one unit, integration
raises the degree by one unit, but neither operation affects the order.
Operating on (1) with A> 'we have
(6) S A = (e* - !)/„(<) e»‘+»(‘).
K SKS 0
Operating on (1) with V. we have
126 THE POLYNOMIALS OF BEBNOULLI AND EULER [6-01
6-01. The p Polynomials. Formula 6-0 (6) suggests that a
particularly simple class of ^ polynomials should arise if we take
in 6-0 (1)
where n is any integer positive, negative, or zero.
The polynomials which arise in this way we shall call p poly-
nomials, and we write
(1)
(e‘-l)«
oxt+g(t) .
>•--0 V!
SO that from 6'0 (6),
V . y =^0 ^ •
whence we obtain
(2) = vp<«_V>(a;).
Thus the operator A depresses both the order and the degree by
one unit.
With the aid of 6*0 (3), (2) can be written in the form
(3) 1)" - + a;)*' = V 4-
Writing x = 0, we have the symbolic equation
(4) 4- 1)- - P<"> = V ptAl”,
which gives a recurrence relation between the [3 numberH of orders
n and n - 1.
6*1. Definition of Bernoulli's Polynomials. The ^ poly-
nonaials of order zero have the generating function The
simplest polynomials of this type are obtained by putting g(t) = 0.
The generating function then becomes and the corrtvspondhig
P polynomials of zero order are simply the successive powers of x.
It is convenient to regard these simplest (3 polynomials as Bernoulli's
polynomials of order zero. We therefore make the following
definition :
BermulWs polynomial of order zero and degree v is given by the
relation
= x\
8-1] THE POLYNOMIALS OF BERNOULLI AND EULER
Thus we have
127
(1)
Then, in accordance with 6-01 (1), we have the further definition :
Bernoulli's 'polynomials of order n are given hy the identity
If we put a: = 0, we have for Bemoulh’s numbers of order n
(3)
£"
From this we obtain
BW = 1, = j5r= ,V»»(3«-1). Bf = -W{n-l),
B^i^ = v,-Jo-n(15w®-30n®+5w+2),
Bf = - 1) (3w2-7w-2),
= ,oViW(63n5-315# + 316n3 + 91w2-42»-16).
6*11. Fundamental Properties of Bernoulli’s Poly-
nomials. Bernoulli’s polynomials are p polynomials and there-
fore also <f) polynomials. Hence we have
(1) = +
(2) =
(3) £ dt =
(4) A5l”>(a;) =
(5) (BW + 1)' - Bl”’ = V Bi’Ll^'.
The first three properties are shared by all <f> polynomials, the last
two hy all p polynomials.
By repeated application of (4), we have, if v > m,
ABi"’(cc) = v(v-l)(v-2) ... (v-n+l)a;''-".
(6)
128 THE POLYNOMIALS OF BERNOULLI AND EULER [G-u
since If v <n, the right-hand member vanishes. Rela-
tions (4)' and (5) form the point of departure of Norlimd’s theory of
these polynomials.
We also note the useful relations derived from (4),
(7) + v
(8) JS:(l) = Sl”4vRiri'\
From (3) and (4), we have
and in particular
(10) rjS<'‘>WcZ< = i5l"-”.
6*2. The Complementary Argument Theorem. The argu-
ments X and n-x are called complementary. We shall now
prove that
(1) B^^\n-x) =
We have, from 6*1 (2),
2
V =0
(e'-D-
r<r*‘
{l~e-r
- (e-'Il)n ^
(-0
v!
whence by equating coefficients of V we have the required result.
This IS the complementary argument theorem. The theorem is true
for any |3 polynomial in whose generating function, 6-01 (1), g{i) is
an even function.
If in (1) we put a; = 0, v = 2^^, we have
(2) lS<:>(n) = 5^.
Thus (x) has zeros at a: = ^, x = 0.
Again with k = |n, v = 2[H- 1, we have
(3)
= 0.
6-3] THE POLYNOMIALS OF BEKNOULLI AND EULER 129
6*3. The Relation between Polynomials of Successive
Orders. We have
,, — n ^ •
y=0
tngxt
If we differentiate both sides with respect to t and then multiply
by t, we have
.'tKv - f) ! ■' ^ ’ (e‘ - 1)« ^ ~ le^iy+i
><-0 • V=0 • V=0
Equate coefl&cients of Then
(1) V j5l'‘\a:) = W-B'“’(a:) + a:vJ5l’l^i(x)-w£j,“+”(a:+l).
From 6-11 (7),
(K+l).
Thus we have
(2) 5<"+^>(x) = (l-^) j5W(x) + vg-l)£« (x).
which is the required relation between Bernoulli’s polynomials of
orders n and n-i-1.
Putting a; = 0, we have
(3) 5l" + ”= (l-;^)Bt">-vl5S,"2i.
Again from (1), putting a; = 0, we obtain
or, writing n+v for n,
(4)
j5(n+v+l)(i)
^ ^(n+v)
n+ V
6-4. Relation of Bernoulli’s Polynomials to Factorials.
In 6*3 (2) put V = n. Then
£^»+i)(a;) = (a;-«)B<r2x(a:) = {x-n) {x-n+l) = ...
= (®-n) {x-n+ 1) ... {x-2) {x-
130 THE POLYNOMIALS OP BBRNOIMA^I AND KULEK [tj-i
Thus
(1) = (a;-l) (x-2) ... (x-n) ^ (x- !)<«>,
(2) (33 + 1) = a:(x-l)(a:-2)...
Integrating these expressions from 0 to 1, we have from fl-l L (10),
(3) f (x-l)(x-2) ...{x-n)dx =
Jo
(4) ^\{x-l){x-2)...(x-n + l)dx =
from 6*3 (4), putting n = -1.
If we differentiate (1) v times (n > v), vve have, using G- l 1 (2),
«(w-i)...(w-K+v+i)Br”w =
which gives an explicit expression for namely,
(5) 1) (a; - 2) ... (x-n) j .
The following coefficients appear in Stirling’s and Besscd’s inter-
polation formulae (3*3, 3*4),
%+i(p) = (2^,+l).
‘=.w7"T‘)'
From (1), we have
®2s+i ip) = (27+ ly t 2»+'r^ ip+s+i).
«2.(f) = (2^)1 ^4"'-l(P + s),
hs+i ~ (2s + 1 ) !
^2«(y) = (2j.),i4r‘'(?)+«)
If we differentiate each of these m times with respect to p and then
put ^ = 0 in the first two and j? = J in the second two, we have
(6) 1).
(7)
6-4] THE POLYNOMIALS OF BERNOULLI AND EULER 131
(8) i>2s+i (1) = (2^i)(2s-m+l)! + i)'
(9)
From these we have, with the aid of 6-2 (3),
(10) a2,+x (0) = 0, i)2>»+i a,, (0) = 0.
(11) = 0, hM = 0.
6‘401. The Integral of the Factorial. A function which
is of importance in the theory of numerical integration is
(1) X(^)=r (y-l)(y-2)-(2/-2«+l)d2/
J 1 — fe
2n
where h is zero or unity. From the complementary argument
theorem of 6-2 we have at once
(2) X(2^ + *-l) = X(l“^) = 0.
From (2) we have, integrating by parts,
r2?i+&--l C2w+A;-1
(3) = “ x{x-l) .,.{x-2n+l)dx
Ji— i J 1— &
Si2«+i)(2_fc) + B«"+»(l-fc)
2w+l
because B'i^^i'^i:2n + k) = - J5^^+Y^(l -*) from the complementary
argument theorem.
Again, = f (2/^2) ... (j/-2w + l)(Z^,
J 0
and it is clear that the integrand is negative when 0<y<l.
Thus is negative. Similarly
= £(«/-!) (y-2) ...{y-‘2M+l)dy,
and we have Bfn-x (1) positive. Proceeding in this way we see that
if V be an integer, 0 ^ v 2w,
(4) (-l)''+’-Bfc“(v)>0.
132 THE POLYNOMIALS OP BEBNOULLI AND EULEK [O-ioi
We now prove that
(6) 1 < w- l.
We have, from 6-11(9),
l«=iW = j ^ (y-1) — (y-2«+i)%
= r y(y-i)--(y-2« + ‘3)%-
Jv-1
(7) = I 1)%.
Now yl{2n-y-l) is positive and less than unity provided that
y <n-\^ which, is satisfied since v - 1 < y < v, and v < n - - 1.
Comparing then the integrands of (6) and (7), we see that th(‘
integrand of (7) is less in absolute value than the absolute value
of the integrand of (6). The result (5) therefore follows.
We can now prove that defined by (1), has a fixed sign for
1 -lb < a; < 2n‘\‘h- 1.
Let X lie between the integers v -■ 1, v.
For v-1 < y < V the integrand of (1) does not change sign and
hence x{^) between the following pair of integrals :
f (y~-l)...(^-2n+l)d!y, f (y- 1) ... f \)(hj,
h-h
If we divide the ranges of integration into intervals
we see that x(i») lies between the sums
-A:) + Sgrrx«(2 -i) + ... -f
Bfc»(l - fc)-f - i)+ ... + Bgr:V>(v - 1).
Here we can suppose that v < n, for by the complementary argu-
ment theorem such terms as exist when v > n cancel out. By
(5) the terms in these sums are in descending order of absolute
magnitude and alternate in sign.
6*401] THE POLYNOMIALS OF BEENOULLI AND EULER 133
Hence each sum has the sign of the first term, namely, the sign
of
Thus we have proved, in particular, that has no
zeros in the interval 0 < a: < 2m, and that
(8) ;i:=:0orl.
6*41. Expansion of x<"> in Powers of x. Differentiating
6*4 (2) f times, we tave
Hv
Putting X = 0, we have
r
n\
0
-'wU v-*-/ “■
V j>(n)
{n-p)ln
from 6-3 (4).
Thus, developing a:<") by Maclaurin’s Theorem, we have
n
V
x^
nl
_P -Din)
^opHn-p)\n
6’42. Expansion of x” in Factorials. We have by Newton’s
Interpolation formula 3T (4), since B^J'^x + h) is a polynomial of
degree v,
(1) (x+h) = B^:^\h) + 2 A 5«(A)
using 6-11 (4). Putting A = 0, we have a factorial series for Bi^^(x),
namely,
(2) BW(a;) = S
Putting n = 0, we have Bf\x) = x% so that
(3)
S==0
which is the required expansion.
134 THE POLYNOMIALS OP BERNOULLI AND EULER [0-42
If we operate on (3) with Ai Put a: = 0, we
obtain the differences of zero (see 2-53), namely,
If in (1) we put n = v+ 1, and h+ 1 for h, we liavc, using C)-4 (2),
{x + hY^^ =
which is Vandermonde’s theorem in factorials analogous to the
Binomial Theorem
From (1) we have also, interchanging x and A,
4”^(a;+A)-.By(a;) ^ ^ /vN («_-») (3.) _
fi \ S/
If we let A — >0, the left-hand side becomes the dcrivate of
that is, V jB i (rjj) , Thus
In particular, for x ^ 0,
> -X = S 0 ( - 1)'"' (^ - 1 ) ! -S l"-V'
6-43. Generating Functions of Bernoulli’s Numbers.
We have, by the Binomial Theorem,
Differentiate n times with respect to x and wc obtain
(l + <)-‘[log(l+I)l" = 2
6-43] THE POLYNOMIALS OP BERNOULLI AND EULER
Putting x=l and dividing by we have
135
(1)
s
using 6*3 (4).
In particular, for n — 1,
(2) log (1 + 0
■^0 v! n + v
Again, integrating (l + i)*-i with respect to x from x to cc+l,
n times in succession, we have from 6*11 (9),
V —0 ^ •
[log (! + «)]'>
Putting a; = 0, we have
t”
(l + f)[log(T+«)|
and in particular, for = 1,
„ ___ ^
\n ~ Zj ^ V 3
«0 V
(4)
(1 + 0 log (1 + 0 v! ’
which is the generating function of the numbers
Again putting a; = 1, we have
(5)
Llog(i + oJ ~M^! ^
which shews that (1) also holds when n is negative.
In particular, for 9^ = 1,
(6)
t
log(l + 0
OO fv
X'-A t'
V
which is the generating function of the numbers
Using 6*3 (4), we have from (6),
t
(7)
W fy 25(v-l)
136 THE POLYNOMIALS OP BERNOULLI AND EULER [6-43
We give a list of ten of the numbers
7:?(<>) 1 1) 0 8 7
x>(5 — j
Bf = 1,
B^P
Bf = -h
11
Bp = - •>
Bf = - YtS
=iaA2.i,i s -s s
6*6. Bernoulli’s Polynomials of the First Order. We
write By (x) instead of the order unity being understood.
Thus from 6'1 (2), we have
(1)
i'
7!
as the generating function of the polynomials and
as the generating function of Bernoulli’s numbers, of the first
order.
From 6*11, we have the following properties :
(3) B,{x) = {B + xy,
(4) = v = 2,3,4,.,..
(5) ^By[x) =
(7)
(8) (1 - a:) = (-!)>' 5, {x), from 6-2.
The first seven polynomials are given in the following list :
■®o(®) ~
BAx) = x-\,
BAx) = 2^-x+l,
6-5] THE POLYNOMIALS OF BERNOULLI AND EULER
Bs{x) = a;(a;-l)(a:--|-) = a;2 + -|a:,
B^(x} ^ 0li^-23^ + 3^--s\,
5j(a;) = x{x-l)(x-l){3^-x-l) = a:^--^x^ + ^^a^-lx,
Bg{x) = a:®-3ic®+-|ic*-^a;2^_t^
We have also for the values of the first seven numbers ;
B^ B, B3 B4 Bj B*
-i 0 --3V 0 ,1-
6‘501. A Summation Problem. To evaluate ^ s".
We have by 6-5 (6) and (7),
fs+l 1
5„(a:)da; = ~[B..,i(s + l)-B..,,(s)]
Thus ” ^ ■
£ 5^ - f " ' B, {X) dx = ~ [B.^i (n + 1 ) - B,,. J.
g = l Jo V+i
For example, if v = 3,
= H(^i+l)^-2(w+l)?+(« + l)®]
= [^n{n + l)Y.
The method can clearly be applied if the 5th term of a finite series
be a polynomial in 5.
6*61. Bernoulirs Numbers of the First Order.
from 6*5 (2),
(1)
t j, e*+l
2 + ^4^^ “ 2 * e^-1*
We have
The function on the right is even, since the change of -t fox t
leaves the function unaltered. It follows that the expansion con-
tains no odd powers of t, and hence
•S2/x+i = 0, ii>0,
Bi = -h
138 THE POLYNOMIALS OF BEI^NOULTA AND EULEIi
If in (1) we write for t, we obtain
2^
t cotb ^ = 1 + -t-
and writing it for t, we get
2“
(2) t cot i =r: 1 - - JS2 +
2**^4
4!
4!
[O-al
Expansions for cosec t and tan t are easily obtained L)y use; of the
identities
cosec t = cot - cot t,
tan t = cot t-2 cot 2t
00 92i-0)2»_n
r « 1
Again we have the expansion in partial fractions/^
(3)
TZt cot 7T^ = 1 -h 2t^
1 '
This series may be rearranged and thus, comparing (5()etri(rK‘rit.s of
in (3) and the series for nt cot izt derived from (2), we have
(^)
^ 2(22))!
00
1
*
The sum of the series on the right lies Ixd-wtHai 1 and 2. Thus
we see that increases rapidly as f increases and t imt, I'>(*rnoullhs
numbers alternate in sign. Moreover, we ha\H^
( -1)^-1 5,, >0.
To express Bernoulli's numbers by detcniunants vvt^ liavt* from
6*5 (4),
= 0
21 1! '
3! 21 ll^i! 2l
0,
1 A
1 B,
(n + 1) ! nl 1 ! (?i -T)l 2 ! ■*' ‘
.+
1 3 I
2! [n - ])!''■«!
* K. Knopp, Theory of Infinite Series, (London, ll)2KU § 1 17, Li, 7.
C-51] THE POLYNOMIALS OF BERNOULLI AND EULER 139
whence, solving these equations, we have for {-ly B^j n\ the
determinant
1
2'!
1
0
0
. 0
1
3!
1
2l
1
0
. . 0
1
4!
1
3!
1
2!
1
. . 0
1
1
1
1
1
(n + l)!
n\
(n-1)! (w
,-2)! ■
■ * 2!
Since Bz^+i = 0,
p > 0, we have
B, {x)-t\'ix'
-1=:
{x + By->r\'ix''-^
= a;-' + ya;>' j54+... ,
so that B, (x) + I'V x''-^
is an even function when v is even and an odd function when v is odd.
6*511. The Euler-Maclaurin Theorem for Polynomials.
Let P{x) be a polynomial of degree n.
From 6*5 (7), (3), we have
It follows from this result that
(1) F{x):^P{x+B + l)-~P{x + B),
and consequently that
(2) P'{x-i~y) = P{x+y + B + l)-P{x + y+B)
^P(x+l^B(7j))-P{x + B(y)).
Now by Taylor’s Theorem
Pix+B(y))^P{x) + B^ (y) P' (*) + B^ {y) P" {x) + ...
140 THE POLYNOMIALS OF BEKNOULLI AND EITLER [0.511
Thus substituting in (2), we have
(3) P'{x+y)=^^P {x) + B, iy) A P' {x) t- , B., (//) A P” (x) i...
This is the Euler-Maclauriii Theorem for a poIynoiniaJ.
In particular, putting ?/ = 0,
(4) P' (x) = AP (^) + B,AP' (x) + B, A (X)
+l,B,AP<‘''Hxri^!,B,AP^'-<Hx)i-...,
since Bg, B^, B„ ... all vanish.
If we now write
P{x)=?‘cl>(t)d/,
J a
we have
(5) <j> (x) = <!> it) dt+B,A<f> (x) + j, B. A {x)
J a? •
•I ,,V/>“,Af"M I'....
Since = - 1, we can also write
fOJ+l 1
(0 dt = i [<{> {x+i) + ci> {X) ] - Ik A <(>' (*r)
X ^ 1
-lB,A<l>"'ix)~...,
where ^(x) is any polynomial. The series on th<^ right; of course
terminates after a finite number of terms.
Again, (1) shews that the difference equation
Au{x) = P'(x)
has the polynomial solution
u(x)=P(x+B) =^P(B (x)).
Thus, for example,
A«(«) = x®-3x^+I
6-511] THE POLYNOMIALS OP BERNOULLI AND EULER 141
has the solution
u (x) = (x) - £3 (x) + j5i {x) + c,
where c is an arbitrary constant. To obtain the general solution, we
replace c by an arbitrary periodic function m{x), such that
tu(5c+l) = w(x).
6*52. The Multiplication Theorem, If m be a positive
integer, we have from 6-5 (1)
m-l /
S bJ:
8=0
mJ
if -0
Thus
t mx' ~
m • — e ^
m
L
e^-1
v!
By (mx).
By {mx) =
This result is known as the multiplication theorem for Bernoulli’s
polynomials of order unity.
Putting a; = 0, we have
Hence, if m = 2,
BAl)=-{l-^^B„ v = l,2,....
6*53. Bernoulli’s Polynomials in the Interval (0, 1).
From 6-5 (8), we have
(1)
(2) •®2i/+l (1 - a?) = “ ^2v+l (^) •
Thus B^y{x)-B^y has the zeros 0 and 1. We shall prove that
these are the only zeros in the interval 0 a; 1.
142 THE POLYNOMIALS OF BERNOULLI AND EIJLEE [6-53
Again, = "" (a’)^ ^2v+iii) = is
sjonmetrical about ic = -I (from (2)), so that i^2K i i(‘^05 '^ < 0,
has the zeros 0, 1. We shall prove tliat these are the only
zeros in the interval 0 < a; 1.
For suppose that both these statements are true uj) to and in-
cluding V = fji > 0. Since
(3) - (2[X-i-2).B2. ! l(^)
which vanishes at a; = 0, a* = 1, has its only
maximum or minimum for 0 < a; < 1 at x =:^ J, and consequently
cannot vanish in this interval,
sain,
= (2[x -f 3) [B2^ I ~ Bofi 1 2] + ("d[x f 3) 1 2 ?
and this expression can vanish at most once for 0 ■< x < h .
Hence S2/x+3(i^) cannot vanish in 0 < x < A and tluu'efore by
(2) cannot vanish in i < x < 1.
By induction the properties therefore follow.
From 6*51 we have (~ 1)^'+’- > 0. If x be suilicuuitly small
and positive ( - 1)^+^ B^^+iix) has the same sign as the derivatc, that
is, the same sign as (-- 1)*'+^ jB2v(^)» which for x small and positive
has the same sign as ( ~ 1)"'+^ which is positive. Thus
(-1)^+1 Bg.+i (x) >0, 0 < a: < .
Hence, from (3), (- - -^2,. i u) iuen\-mes from the
value 0 as a:; increases from 0 to i and is thenilbre positiv(*. Hence
( - i)/^ (x) - J?2m) > 0, 0 < a; < 1 ,
since the expression only vanishes at 0 and L
6*6. The 7] Polynomials. A second method of generalising
polynomials is suggested by 6*0 (7). If we write /„ (/-) - 2'*’ {e^ l I )
.we have a class of polynomials, which we call 7] polynomials, given by
(1)
so that
(e‘ + l)"
v-o
v!
6-6J THE POLYNOMIALS OF BERNOULLI AND EULER 143
whence we obtain
(2) =
Thus the operator V depresses the order by one unit but leaves
the degree unchanged.
Using 6*0 (3), we have
(3) (t] + ic + 1 ) " -I- (t] + a;) 4= 2 (t] + x) \
so that the t] numbers satisfy the recurrence relation
(4) (t) + 1 ) *' + 7] r“ 27] ~
6‘7. Definition of Euler’s Polynomials. The simplest
7) polynomials, obtained by putting = 0, n ~ Q in the gener-
ating function, are the powers of x, whose generating function
is We shall now regard these simplest t] polynomials as Euler’s
polynomials of order zero. Thus
where {x) denotes Euler’s polynomial of order zero and degree v.
Then in accordance with 6*6 (1) we define Euler’s polynomials of
order n by the relation
(1)
'')npxt 00 fv
In accordance with our general theory we should call Euler’s
numbers the values of (0). This would, however, run counter to
the notation of Norlund, who discovered these generalised poly-
nomials. In order therefore to avoid confusion with the accepted
notation we shall follow Norlund and write
(2) = 2->'<7l”\
The generating function for the 0 numbers is therefore
(3)
2« _ 1
Gf>.
The values of for x = ln are called Euler’s numbers
of order n. Thus
(4)
We shall prove in 6-72 that Euler’s numbers with an odd suffix all
vanish.
144 THE POLYNOMIALS OP BERNOULLI AND EULER [6-71
6*71. Fundamental Properties of Euler’s Polynomials.
Euler’s polynomials are t] polynomials and therefore also ^ poly-
nomials. Hence we have
(1) +
(2) =
(3) £ {t)dt = [^<1 1 {X) - ,(a) .
(4) =
from 6-6 (4) and 6-7 (2) Thus
(5) (G« + 2)>' + Cl,”>^2C'l”-‘\
By repeated application of (4), we have
(6) . 'S/ E^”\x) = x",
since Ef^ (x) = x”.
We have also from (4),
(7) = 2E^^-'^\x)- E^^\x).
Since = 2-E<!‘>(|) =(|G('»+2)\ 2- from (I),
we have
Hence we have
(8)
£(«)=: n+C<">.
Thus we have, by putting in turn a; = 1, a; ==■ - 1 , and adding
(£(«)+ l).+(£(n)_ 1). 4: +
= 2>'+ivB£.”>(-2
4- 2 -+1 J! (« - 1) zJ- j from (4 )
= 2E’ir-'^\
(9)
6-72] THE POLYNOMIALS OF BEBNOULLI AND EULER 145
6*72. The Complementary Argument Theorem. The
arguments x and n-x are called complementary. We shall now
prove that
(1) B^f>{n-x) = {-VYE^:'\x).
We have from 6-7 (1),
2« (>{n~x) t
On p-xt
(e'+T)" (c-‘+T)'‘
whence hj equating coefficients of we liave the required result.
This is the complementary argument theorem, Tlie thconmi is true
for any 7} polynomial in whose generating function g{t) is an even
function.
If in (1) we put x = 0, we have, for v = 2p.,
EfJUn) = Ef^{0) =
Thus i^) - 2 has zeros at x = 0, a; = 7i.
Again, putting x = I'-w, v = 2[x-\- 1, in (1), we have
Tp(n) pin)
I ~ “ 2/a -1-1 •
Thus £"2^+1 = 0, that is, Euler’s numliers with a,n odd suffix are
all zero.
6*73. Euler’s Polynomials of Successive Orders. We
have
M fp On pxt
(c'+I)«*
Differentiate both sides with respect to t and then multiply by t.
We then have
+1
Equate coefheients of i’'+^. Then
E<irli (x) = X (x) - 1).
146 THE POLYNOMIALS OF BERNOULLI AND EULER [(iUli
Now by 6-71 (7).
Ef+^\x+l) =
Therefore we have the recurrence relation
Writing a: = 0, we obtain
^(n + l) — ± 0(ti) ^ 2
6‘8. Euler’s Polynomials of the First Order. We
stall write Ey(x) for Ei^^x), the order unity being understood.
We have then frona 6-7, 6-71
(1)
(2)
2e=“
2eU
e*+l
r 1
v! 2^
o i-
^ v! 2*'
Cy
> 0.
(3) Ey{x) = {\C + xY, {G+2Y+Cy-^(\
(4) '^Eyix) = x”, DEy(x) = v£!,_i(a:).
(5) Ey{l-x) = (-1Y Ey{x), from G-72.
The first seven polynomials are given in the following li.st :
Eoix) = 1,
E\Y^') — X j>,
E^{x) = x{x-l),
Es{x) = 1),
E^{x) = x(x- l)(;c“-a:- 1),
E5 (x) = (x- 1) (a^ - 2a;^ - X“ + 2x ■ I 1 ),
Ee{x) = a:(a:- l)(a^- 23,-® - 2a;® + 3a; | 3),
^2 E^ E, E^ Eri
-1 5 -61 1385 -50521 2702765
6*8] THE POLYNOMIALS OP BERNOULLI AND EULER 147
n
Example. Evaluate
a = l
We have by (4),
8=1 8=1
^1^{-IY\eAs + 1)+E.{s)\
s==l ^ J
6*81. Euler's Numbers of the First Order. From
6*8 (2), writing 2t for t, we have
Thus
00 iv
sech t
and writing it for
(1) sec<= + .
Again, by rearranging the expansion,*
4 cos
(-ir(2v + l)
(2v+ l)2-x2 ’
and equating the coefficient of to the coefficient of in the
7lX
series for Jtt: sec -g- obtained from (1), we have
~2p+i _ 1 J: L ^
I ''' 0 9.01.1.1 I
1
323)+1^52p+1 723JH-1
+ ... ,
22p+2(2^)!
which shews how Euler’s numbers increase and that they alternate
in sign.
* See K. Khopp, loc. cit p. 138.
148 THE POLYNOMIALS OF BERNOULLI AND KULER [O-Sl
By tte mettod used in 6-51 we obtain from the recurrence relation
a determinant for (2n)\, namely,
_1 1
4l 2!
2 1
6! 4!
0
1 0
1 1 1
(2w)! (2n-2)! (2n-4)! (2m- (5)!
With regard to the numbers C^, we have from C-8 ( 1)
00 -fv
v! 2*'
vs«0
1
= -tanhif,,
wMch is an odd function, so that all the numbers (i > 0,
vanish. Writing 2t for t, we have
tanh t — -g| C5 “ ^ I C7 — ... ,
whence with it for we get
/3 fS f
tan^ = + ••• •
5!
7!
If w^e equate corresponding coefficients in this series and the series
for tan t in 6-51, we have
0.
2p”’1
22v(22v_1)
2^;
Since the numbers (a > 0, all vanish, wc note that
Ey{x)-x^ =
(7j+
3/ 2^
O3+
j
so that By {x) - a:' is an odd function when v is even, and an even
function when v is odd.
6*82] THE POLYNOMIALS OP BERNOULLI AND EULER 149
6*82. Boole’s Theorem for Polynomials. From 6-71 (8),
when n = 1, we have
E^{x) = {x+lE-^iy.
Hence
2x^ = 2 \7 Ey (x) = (aj + 1 + (07 + — •g)*',
and if P{x) be a polynomial,
(1) 2P(a^) = P(it;+l + -p~-|) + P(x+P-i).
Writing x + y for x,
2P{x + y) == P{x + y + l + ^E-l)-\-P{x+y + ^E-^)
^ P{x-i-l+E (y)) + P{x + E {y)).
Now, by Taylor’s Theorem,
P(x + E{y)) = P{x) + iy) P' {x) + ~ E,^ {y) P" (*) + ...,
Thus we have
(2) P{x + y)=^^P {X) + E^ (y) V P' {x) + E^ (y) ^P"{x) + ...,
which is Boole’s Theorem. If we put a; = 0, we have the expansion
of P{y) in terms of Euler’s polynomials.
From 6-72, we have
E,,{l) = E,M = ^-^^C,, = 0,
Hence putting ?/ = 1 in (2), we have
(3) P{x+l) - P{x) =-C,V P'{^) - (?3 V P'"{^)
-^,G,VP<~-\x)-....
Again from (1), we see that the difference equation
yu(x) = P(x)
has the solution
u(x) = P(x + ^E-^) = P(E(x)).
Thus, for example, the equation
\/u(x) = x^+2x^+l
150 THE POLYNOMIALS OF BERNOULLI ANB EULER lo-sa
ias the solution
u(x) = £'3(a:) + 2^2(a-) + l.
The general solution may be obtained by adding to this an
arbitrary periodic function 7c{x) of period 2, and such that
7r(a:+ 1) = -7T(a;).
EXAMPLES VI
1. Prove that
(i) + = [B(”‘>ix) + B<”>(yyr ;
(ii) B^:^ (ct + y) - V Q JB<-' (X) y'-^ ;
^«s0 -r
(iii) Q) B,
2. Obtain the formulae
(i) {x + yY = Bf\x) B\Zli{y) ■
(ii) i^^yy=tQEf\x)E\r4yy)-,
(ii)
{x +
(iii)
t
p«=0
3. Prove that
,11 1
1 + O + o + ... + - = 1 )
4. Prove that
5. Prove that
E^r^^\x + y) = + ^
6. If P(a;) be a polynomial, prove that
P(5(«) (CC) + 1) - P(P(n) P'
BX. VI] THE POLYNOMIALS OF BERNOULLI AND EULER 151
and hence shew that the difference equation
is satisfied by
(a;)).
7. Prove that
Bf\x) = x-1, B^?\x) = a:^-2a: + |-.
8. Draw graphs showing the forms in the interval 0 < cc ^ 1 of
(-lyE^A^y
9. Prove that
2 2n+l
P A , , A 02^+1
^2«-- fa
10. If n be an odd integer, prove that, taking \{n- 1) terms,
11. Shew that
12. Prove the relation
(T«jl~2! ~ (4n-4)! 6! + + (4^":r2)r
13. Obtain as a definite integral from the identity
e* + l 2
, f® sina;i
14. Prove that the coefficient of 0^^ in the expansion of (0 cosec Oy
is 2^^(2n-l)(--l)^~^B2^/(2n)L
15. Shew that the coefficient of z'^ j n\ in
I log (1 ~ e~^) dt-z log z is numerically equal to — .
Jo i
16. By means of Bernoulli’s numbers or otherwise, prove that
P 22 32 _ 2tz
12+1*22+1*32 + 1
[KX. VI
152 THE POLYNOMIALS OF BEKNOULLX AND .PNJLNR
17. Prove that
1
2!
7t‘*
4!
^4-
0.
18. Express the sums of the powers of numbers less than n and
prime to it in series involving Bernoulli’s numbers.
19. Shew that
1-1 ^
3^'^ 5^
1530 ■
20. If S"= 1"+2”h-3”+...4-x’*, shew that
rsni
21. If ^(2;) = l”H-2”4-3”-h... + (a;- 1)^*, shew that I\x) is a poly-
nomial in (x — ^) and cannot contain both odd and even powers of
the same.
22. Prove that {o+(n-r)}«+»- expresses the sum of all the
homogeneous products of 5 dimensions wliich can be formed of the
r-hl consecutive numbers n, n- 1, , n-^r.
23. Express x x(^> in factorials.
24. If denote the number of combinations of m things r
together with repetitions, and if 0^ denote the number of com-
binations of m things r together without repetitions, then
^r=-,Ao™+-
ml
and 0? is obtained by writing -(m + 1) for m in the expanded
expression for iff.
25. Prove that
^ 2 M+1
26. Expand (x~ !)(-«) in powers of x-K
EX. VI] THE POLYNOMIALS OP BERNOULLI AND EULER 153
27. Expand in factorials of the form (a;
28. Prove that
(a? + 1 ) = f 2 (cos {\z) )® cos {\xz) cos mz dz.
7C J Q
29. If Di = x^j shew that
2
f{x) being a polynomial.
n / 1 \m w
30. Prove that jB„ = V L \ Qn^
31. Prove Staudt’s Theorem, namely, that every Bernoulli
number is equal to an integer diminished by the sum of the
reciprocals of all and only those prime numbers which, when
diminished by unity, are divisors of 2n.
32. Prove that 2 ~ ^ 1.
w-s+1 s!
33. Prove that
g(»+.+2)(i) ^
CHAPTER VII
NUMERICAL DIFFERENTIATION AND INTEGRATION
The problem of numerical differentiation consists in iinding an
approximate value of the derivate of a given order from the values
of the function at given isolated arguments. The |>rol)lem of
numerical integration consists in finding approximately, from the
same data, the integral of the given function between definite limits.
In this chapter we shall investigate a few of the many formulae
which have been proposed for this purpose. It will be .t'ouiul that
the generalised numbers of Bernoulli enable us to ol)tain gtuieral
expressions for the coefiicients of most of the formulae, it may l)e
observed that some of the methods of numerical inti‘gral ioii (often
called mechanical quadrature) lead to corresponding methods of
summation when the integral is known.
7*0. The First Order Derivate. We hav(‘ from N(*wk)ifis
formula 3T (4),
f(x+y) = fix) +pA m + - + {J_ J zl" -VW + il)
where <o denotes the tabular interval and p = y j oi.
Thus
/(a^+y)-/N
= Af{x) +
(23-1)
ZlV(*) +
3!
zlVW
+ Ji- <!!z» 1 3) J-,,.)
+(£zl)^pi±il
7-0] NUMEBICAL DIFPEKENTIATION AND INTEGRATION 155
If we let y^O, then ^)->0, and
whicli expresses /' (a?) in terms of the differences of /(a;).
To use the formula we have therefore to form a difference table
in which x figures as one of the arguments.
The above method can of course be applied to any of the inter-
polation formulae of Chapter III. .Thus, from Newton’s backward
formula,
CO / (X) = zj /(^ - CO) + - 2<o) -h fix - 3co) + . . .
Again, if we use Stirling’s formula we note that the coefdcients of
the even differences vanish when p->0, so that from 3-3 (3), for
example, we have
the differences pS Uq, Uq lying on a horizontal line through the
tabular value / (x). See also Ex. VII 21.
These formulae have been obtained by a special artifice which
gives the remainder term in a simple manner. We now proceed to
a more general method,
7*01. Derivates of Higher Order. Let
(j>,ix) = (x-x^) ... (x-a;,).
Then Newton’s formula for interpolation with divided differences
can be written
n— 1
fix) =f{xA+ ^ 4>s {x) [x^x^s . . . + 4 „ix) [xx^x^ ...x„].
s--^l
Differentiate m times with respect to x. Then
(1) Z; <^f'^{x)[XjX^...x,+j} + RAx),
(2) where R^x) = ^{(j>„ix) [xx^x^ ... »„]},
which expresses the mth derivate in terms of divided differences.
156 NXJMEEICAL DIFFERENTIATION AND I NTEO RATION [7-01
To deal with, the remainder term, let us first suppose that x is
not interior to the smallest interval I which contains x^, x.>, , x„.
Since has n zeros all in I, by repeated application of Rolle’s
Theorem we see that if m < n, has exactly n - m zeros all
in I. Hence, if t/ be a point exterior to I, f 0.
Now consider the function
This function vanishes for x = x^, x^, ... , a;„.
Let J be the smallest interval which contains y, Xj, x.,, ... ,
Then, hy repeated appheation of Rolle’s Theorem, (x) has at
least n-m zeros in 1, and also the zero y which is not in 1. Thus
has at least n-m+l zeros in J. In particular, if m = n,
then has at least one zero, say ■>], in J. Thus
d”_
dv]”
Now, from (1),
Thus
(4) Hence
ii>r^ (^) h ^1^2- *n] } = (^)-
^SrMy) n'-
B„ix) = i^^ci>ir\x),
where t] is some point of the interval bounded by the grcatcist and
least of Xj ccg, ; and a; is not interior to the interval bounded
by the greatest and least of Xj., x^, ... , x^- Of course x may bo an
end-point of the latter interval.
If in the second place we suppose x to be interior to the least
interval I which contains we can proceed as follows.
By Leibniz’ Theorem,
K{«>)
Now by 1-8,
v«0
a"
dx'
= [xx ... xx^x^ ... ®„] (m- V)!,
7-01] NUMERICAL DIFFERENTIATION AND INTEGRATION 157
where the argument x occurs wi - v + 1 times and
where is some point of I. Thus
(6) g
Comparing (4) and (5), we see that, when x is not interior to the
interval J, the form of the remainder term in (1) is obtained by
differentiating m times the remainder term as if ^
were a constant, although the value t) finally used may not coincide
with the original value If x be interior to the interval I we obtain
the more complicated form (5).
7*02. Markoff’s Formula. If by 6-4 (2)
Thus Newton’s interpolation formula can be written
f{x+y) = S + +
where ^ is some point of the least interval containing
(x+y, Xy x+{n-~ l)ci)).
If y =: Oy x+y becomes an end-point of this interval Thus we can
determine the remainder term of the mth derivate when y = 0 by
the formula (4) of the preceding section.
Differentiating m times with respect to y and then putting y = 0,
we have
« =771
+ - n (« - 1) ... (n - m+ 1) (1)/W (r)) .
This is Markoff’s formula. Since, by 6*3 (4),
158 NUMERICAL DIFFERENTIATION AND INTEGRATION [7-02
the formula can be written.
(1) (x)
2
Wten m = 1 we have, by 6*4 (1), Bi^li = ( - 1)^“^ {s-l)\, so that
tbe formula agrees with 7*0 (1), wbicb is a special case. The
coefficients can be calculated with the aid of the table in 6*1.
If we write n = mH- 1, we obtain
<o»/(*»)(a;) = /(«•+!) (yj).
From 6-1 (3), = = -^{m + l) .
/ W [x) = A f{^) - hri' to (y]),
which measures the error committed in replacing a derivate by a
difference quotient.
For the case m = 2, n = 6, (1) gives
(4)
+ 11 A* fix) - 1 JS fix) + [ A ■ CO® /(«> (rj).
Example. To find/'(T60),/"(T60) wben/{x) = cosx.
Using the 10 figure tables in the example of 3T2, we have
(■001)/' (-160) = - •0001598118+ -00000049355 -1-1 (-001)3/"'
In this range/'" (Q = sin ^ = 0-16 approximately.
Hence
/'(•160) = --1598118+ -0004936
= --1593182,
which agrees with - sin -160 to the last digit. The last digit is in
general unreliable since the first difference in a correct table may
be in error by one unit. For the second derivate we have
w3/"(-160) = - -0000009871 - CO*/"' (5),
7-02] NUMERICAL DIFFERENTIATION AND INTEGRATION 159
whence /"(-IGO) = - 0*9873, which disagrees by a unit in the last
place with the correct value to four places, namely
-cos *160 = -0-9872.
Since the last digit of the second difference may be in error by
two units, we cannot in any case rely upon the last digit of the
calculated second derivate.
We also observe that although a 10 figure table has been used we
have only determined/' (x) to seven figures and/" {x) to four. In any
case we cannot obtain more figures of a derivate than there are
digits in the difference of the corresponding order.
To obtain Markoff’s formula for ascending differences we begin
with Newton’s backward formula,
J{x + y) = +
Now by 6*4 (1),
Proceeding as before, we get
(2) »-/<-.(.) = s'
{n-m)\n
The coejB&cients have the same absolute values as the coefficients
in (1).
The simplicity of the remainder term in Markoff’s formulae makes
them often preferable to the central difference formulae which will
now be obtained.
7*03. Derivates from Stirling’s Formula. Writing
p = y I Stirling’s formula 3-3 (1) can be written
w -1 n
f{x + y) =Mo+ S ^
« = 1
160 NUMERICAL DIFFERENTIATION AND INTEGRATION [7-03
■wkere =/(a:+33co) and
= {IX\) . = I (^2^-”/) ■
Diffeientiate 2m times with, respect to y and then put y = 0.
Then by 64 (7), (10), we have
where from 7*01 (6),
(2) Rzn+ii^)
(2n+i)! 2;
^2t.+2n-2.+2(2OT) !/G”»+an-2-+2)
(2v — 1) ! (2m + 29^- — 2v -f- 2) ! (299^ ■— 2v + 2) !
Similarly, if we differentiate (29n+ 1) times, we obtain
(3) co2m+l/(2m+l)(ic)
1
= .ll(2s-2m)!
11) (s + 1) Wo + (:c),
(4) Rzn+li^)
= (299.4* 1) !
0 (2v4■l)!(2m + 2*/^-2v4■l)!(2n-~2v)!
The following list gives the coefficients of the first few terms,
o(n)M ^ n(175#+420w2+404n + 144)
^8 (!«)- 28x3®x5
from wiich we easily obtain, taking 7 interpolation points,
“/' (2;) = [iS ^ 1^53 Mo + -gV [1.5® ^ .
®~560
/<«>(?),
(^) ““ 'Wq 4- -^rV S® W|
7-03] ISrUMEBICAL DIFFERENTIATION AND INTEGRATION 161
7'04. Derivates from Bessel’s Formula. With y j (x> = p,
Bessel’s fonnula, 3-4, can be written
f{x+y) = UQ+pSui+ j] h,+i(p)B^+^Ui+ S ^25(2^)
8 = 1 8 = 1
+ w2’^62„(p)/<2«)(^),
where W3, =/(x+i9 CO ) and
Differentiate 2m times with respect to y and then put «/ = |co.
Then by 6*4 (9), (11), we have
„2m/(2m)(3,+^^) = ”s (25_2.;«yf + + R,„ix),
where by 7*01 (5),
R^{x) ^ {2n)\ ^
,/ = 0
Similarly,
^2 m+iy (2 m+l) (a; -f- |co)
t^2m+2n-2. (2^) !/(2m+2n-2.) + » (w + |)
(2v)! (2m*f 2ri~ 2v) ! (2^- 2v)!
2m+l
(25+1) (25 -2m)
■Sg*J-2»(s+i)82»+iM*+i?2„(a;).
^2»(*)
_/0 U w2'«+2»-2''+l(2m + l)!/(2™+2«-2''+l)(^,)Bg"+2V(w + ^)
(2v)!(2TO+2w-2v + l)!(2ra-2v)!
The formulae for m = 0, m = 1, give respectively
co/'(a=+ico) +
coV"{a;+i6)) =
162 NUMERICAL DIFFERENTIATION AND INTEGRATION [7-04
It will be seen that tbe complicated form of the remainder term
may often render the use of Markoff’s formulae preferable to those
with central differences.
7'05. Differences in Terms of Derivates. By Maclaurin’s
theorem
where ^ is in the interval (0, x).
Now so that by 6-11 (4),
We have therefore
ji - m \‘*’ )
wherej by the method of 7'01, we can prove that
R M - R(— )te)
This formula is due to Markoff. See also 2-54.
7*1 . Numerical Integration. The problem of numerical
integration or mechanical quadrature is that of evaluating
(1) r f{x)dx
Ja
in terms of the values of f{x) for a finite number of arguments
^o> ^2> ••• 3 The methods of approaching this problem fall
into two main groups :
(i) Methods depending explicitly on the values
/(»o). /(ah).- -./(*«)■
(ii) Methods depending on differences or on differential co-
efficients.
We shall deal with each of these groups in turn, but before doing
so we make the general remark that the substitution
_ (b-a)t’ha^-boL
^
7-1] NUMBEICAL DIFFERENTIATION AND INTEGRATION 163
leads to
f* f( \j P r[{b-a)t + a^-bx']b- a
I. = \J L FS*"'
so that the original limits of integration may be replaced by any
others which may happen to be more convenient. In particular,
I f{x)dx—{b-a)^ f[{b--a)t + a\dt,
= £/['
"(6 -a)t + an\
dt,
b~a f+* ^p(6--a)^ + 7c(6 + an
2h
dt.
It follows that a formula established for apparently special cases
such as
rl Cn p + Z;
(j>{x)dx, <^{x)dx, (f>{x)dx,
Jo Jo J -k
can be immediately applied to the general case (1) by a suitable
linear change of variable.
7*101. The Mean Value Theorem. We shall make fre-
quent use of the following theorem.
Letf{x), (j)(x) be integrable functions in the interval {a, b) and let
(j){x) have a fixed sign in this interval. Then, if f{x) be continuous for
a ^ X b, we can find a point ^ in this interval such that
f f{x)<i>{^)dx=f{^) f j>{x)d,x.
J a J a
Let M, m be the greatest and least values oif{x) in the interval
a ^ X and suppose that ^ (x) is positive. Then we have
f [M-f{x)](j>{x)dx'^Q, f [f{x)-rri\j>{x)dx^O.
J a J a
Thus
r6 rb rb
M\ cj){x)dx'^\ f[x)^{x)dx'^m ^[x)dx,
J a J a O'
and hence
f f{x) (f>(x)dx = L j* <l>{x) dx,
J a J a
164 NUMERICAL DIFFERENTIATION AND INTEGRATION [7-101
where Since f{x) is continuous, f{x) attains the value
L for some point ? of the interval (a, b) and therefore L = f{^), which
proves the theorem when ^(ac) is positive. If <ji{x) be negative we
reverse the above signs of inequality and obtain the same result.
7*11. Integration by Lagrange’s Interpolation Formula.
We have, from 14,
(1) /(^) = E S;
(2) = (x-Xj){x-X2)... {x-X„}.
Thus, integrating from a to b,
(3) f f{^) da: = 2
•la i =1
where
(4) , J2„ = £ 9^ (a;) [xx^x^ . . . x„] dx.
Thus the coeflB.cients depend upon the interpolation points
ccj, ccg, but are independent of the particular form of/(x).
Formula (3), like the identity (1) from which the formula arises, is a
pure identity and therefore of general application. The utility is,
however, limited unless an adequate estimate can be made of the
remainder term
Denote by I the interval bounded by the greatest and least of the
numbers a, h, iCg, When x lies in I we have, by 1*2 (2),
[a;aia:ii...a:„] = i/(»)(TQ>,
where t] also lies in 1. Thus if ^{x) have a fixed sign when x is
in {a, h), we have by the mean value theorem, 7*101,
(5) =
where ^ lies in J.
If the sign of <f>{x) be not fixed we can proceed as follows. The
zeros of the polynomial (f>{x) are ... , x^, which we suppose
7*11] NUMERICAL DIFFERENTIATION AND INTEGRATION 165
arranged in order of ascending magnitude. In the interval
<f> {x) has a constant sign and we have
J 1 ^ * J ^s—i
Hence if Xy be the first interpolation point such that a ^ and
Xfj, the last such that x^j, ^ 6, we have
(6) K = <f>ix)dx+...
p ^{x)dx+i^-^^l!^^ j,{x)dx,
JXiL-1 JiCu.
where all lie in Z.
7*12. Equidistant Arguments. If in the formulae of the
last section we take
Xs = a + 5co, 5 = I, 2, 3, ... , n,
we obtain
(1) f f{x)dx= ^ Z,<«)/(a + s<o)
J a s — 1
rb
+ {x- oL-w) ... (x-ix-n(x))[x, a+cxi, ... ,x+n(x)]dx,
Ja
K (») - — ix-x-noi) ,
^ Ja(cc-a~sco)(s- 1)1(71-5)!
Now put
(2) a = a + (l - Jfc)co, b = c(.+ (n + k} co, a? = a+yco
and write F{y) = /(a + yco). We then have
b-a
^~n + 2*-r
= o)(-l)«~«n
5-“l/ Ji-fc V n /y-s
[x, a+w, ... , oc+nco] = ^ /^”>(a+«»7j)
= {0-«[2/, 1, %,...,n\.
(4)
166 NUMEKICAL DIFFERENTIATION AND INTEGRATJON [7-12
Thus
(5) to r f («/)% = to J 1% F (6') + to Rn ,
Jl-& «-l
(6) ]?„= (y-l)...{y-n)[y, I, 2, , n](ly.
Jl-fc
The formula (1) is completely equivalent to (5), but we note that
the remainder term of (1) is
If in (5) we take k = 0, we have
(7) I F(y)dy = + +
while if i = 1, we have
(8) ['^^F{y)dy = Jtl\F{l) + Jt\ F(2)+ F (n) f remainder.
Jo
The essential distinction here is that in (7) the values ^'(1), F{n)
correspond to the end-points of the interval of integration, while in
(8) the values F{1), F{n) correspond to points within the interval of
integration. Steffensen has given the convenient epithets '' closed ”
and ‘‘open” to the first and second of these types of formulae.
Formulae of the open type are useful in the numericail solution of
differential equations where it is necessary to extend the range of
integration beyond the values already calculated. In order to obtain
practical formulae from (5), we must proceed to a discussion of the
remainder term.
7*13. The Remainder Term, n odd. In 7-12 we suppose
that n = 2m - 1 . Then
R.
’27n~l '
= r
+ 1
Ji-.
{y-l)i^-2)...(^-2m+l)[y,l,2,...,2m-l]dy.
Put )'(a:)=f {y-l)...{y-2m+l)dy.
Jl-k
By 6401 (2), 1) = X(^ Hence, integrating
by parts, we have
C2?w4-^•-l
^2m-i = - xiy) [y> y, i. 2, ... , 2m - l]dy,
J 1 -iS
dy
\y, 1, 2, ... , 2m- 1] = [y, y, 1, 2, ... , 2m- 1].
Since
7-13] NUMERICAL DIFFERENTIATION AND INTEGRATION 167
Since xiv) lias a fixed sign (see G-dOl), we have, by use of the
mean value theorem,
[i'll’ 1]
+S-1
1-i
xiy)^y
p2m+r--
(2m)! Ji_i.
ySy)^y
^1]) f p(2w + l) /o j.\ I T7f2m hn/1 7,")
(2m+l)!
by use of 6-401 (3). Thus we have
(1) «2«-l=
where 0 < v] < 2m; and using 6-11 (7),
(2)
(3)
G.
2m~l» 0
9D(2m+l) oo(2m) T>(2m~l)
(2m+l)!’^'(2m)! ■^(2m-l)!
9To(2m+l) T)i2m)
p *-'‘^27^+1 I J^27n
(2m+l)!'^(2m)!’
7*14. The Remainder Term, n even. In 7-12 we now
take n = 2m. Then
R.
r2m+1c
(y- 1) ... (y-2m) [y, 1, 2, ... ,2m]% =
J 2 — Jq
pm+k-l
(2/-l)(2f-2)-..(y-2m)[y, 1, ...,2m]ci:y,
•J 1 — h
:2m =
iy
J
n-k
1
r2m+k-l
if
L.
r2m+l*
^2m+k—l
(2/- 1)(2/'“2) ... {y-2m) [y, 1, ... ,2m]dy.
By the definition of divided differences, we have
(y-2m) [y, 1, ... , 2m] = [y, 1, ... , 2m-l]-[l, 2, ... , 2m].
Si can therefore be expressed as the sum of two integrals of
which, by 6*401 (2), the second vanishes. Thus
r2m-+fc~l
^1= (2/-l)---(2/-2m+l)[y,l,.2,...,2m-l] = iJ2„_i
J
= (52i”VY’(2-^)+5aY^(l-^)}.
168 NUMERICAL DIFFERENTIATION AND INTEGRATION [7-14
Again by the mean value theorem,
Sz = h2> 1. - > 2m] («/- 1) (y-2) ... {y-2m)dy
J2w4-Jfc— 1
= (BgS;'>(2-i)-S|5S«(l -i)),
since Bfm+i\2m + k) = by tbe complementary
argument theorem. Now, by 6*401 (8), the coefficients of the
derivates in and have the same sign. Hence in the sum
^1 + ^2 replace the derivates by a mean value and we
obtain
R2,n=
(1-k),
where 0 < tj < 2m + 1 and
(2) 0 =
(2m+l)t/ (2m)! ’
(3)
(2m + i)! ’
7*2. Cotes’ Formulae. If in 7*12 we put i = 0, we have
h-a
CO =
n- 1
and consequently
? f{x)dx^{b-a) ± /n"7[a-f (v-l)co]hE,„
where
The remainder term is obtained from the formulae of the last
two sections.
We have, with the previous notation.
F<’'){y) = «■'/<*') (a:).
7-2] NUMERICAL DIFFERENTIATION AND INTEGRATION 169
Thus
If, for brevity, we put
(2) 2/v =/[« + (''-
we have Cotes’ formulae, namely,
(3)
f,
b 2m— 1
/(.)*= (6-) S
(Sm-l) ( b-a \2m + l
(*& 2m /n \ / h n \2w+l
(4) \j{x)dx={b-a)^^
Expressions for C q, €2^,0 terms of generalised Bernoulli’s
numbers are given in sections 7-13, 7-14. Numerical values can
be obtained from the table of 6-43.
Cotes’ formulae are of the closed type, the functional values for
the end-points of the range being used in the formulae.
The coefficients have the property
TJ(n) __ TT{n)
y — /I
which expresses that coefficients equidistant from the ends of the
interval are equal. To prove this we have
Siu -
Put t = 1 + - 2:, then
ff<r4+i = (-ir-';
/n-l\
r_Li
/'n-z\
Vv-lA
In 2-v'
K n J
f”J_,
Vv-1/
1 ‘
\ n J
dz
That the coefficients are rational numbers is evident from the
definition.
The values of the coefficients were calculated by Cotes for
n = 2, 3,..., 11.
m
M.T.C.
G
170 NUMERICAL DIFFERENTIATION AND INTEGRATION [7.0
The values in the following table are taken from Pascal’s Reper-
torium. The last column gives the remainder term with the
coefficient abbreviated to two significant figures.
\ V
1
2
3
4
Remainder
2
l_
X
1
3
1
li
4
w
1
0*
-3-5(i>-a)5/to(^)xl0-<
4
i
;}
8
3
8
1
8
- 1-6(6 -a)5/('‘)(^)x 10-4
6
vV
iTxr
1 2
jTu
ll O'
-5-2(b-ayf(o>(^)xlO-^
6
7 fl
'ii 8 F
5 0
28 8
r> 0
2 H 8'
- 3-0 (b-ayf<'‘> (0x10-^
7
yVo
2 16
a 4 (T
!
3 7
HlfW
'K'i'o
-6-4(6-a)»/<®)(5)x 10-10
8
rfil'Tr
3 5 7 7
mwiT
IJi 3 3
iT-iHF
2 V K U
1'7'2 8'cr
1
-4-0(6-a)9/(«)(5)x 10-10
The remaining values for n = 5, 6, 7, 8 are obtained by using the
relatioE = HSTi-.+i-
Comparisoa of the remainder terms shews that there is little to
be gained by taking ordinates instead of 2m- 1.
7-21 . The Trapezoidal Rule. Cotes’ formtila for n-2
gives
£ f{x) dx = |[/(a) +/(6)] - il).
To apply this rule to a given interval (a, b) we may suppose (a, b)
divided into n equal parts, so that b-a — nh, say. To each point of
division, including the end-points, there will correspond a value y
If to each separate part we apply the rule we obtain with
an obvious notation
jjdx= (j/, + J /" (5.))
7-21] NUMERICAL DIFFERENTIATION AND INTEGRATION 171
= K\yi+yi+yi+ —+yn+iyn+i\- il)
= [i-yi+2/2+ — +2/n + i-2/„+i]-^j^/"(?)
which is the trapezoidal rule.
7*22. Simpson *s Rule. This well-known and useful formula
of mechanical quadrature is the special case m = 3 of Cotes’ formula.
We have then
jy (a;) dx = t^ [/(«) + 4/ {^) +/(6)] - /(4) (^).
The remainder term is zero and the formula exact when/(ir) is a
polynomial of the third or lower degree: If we divide the interval
(a, h) into 2w- equal parts, so that 2n}i = 6 - a, we have, applying
Simpson’s rule to each successive pair,
\ ydx = ^[2/1 + 42/2 + 2^3 + 4^4 + ... + 22/3„_i+42/2„+2/2„+i]
•+•72(2^-, A),
where
7'23. Formulae of G. F. Hardy and Weddle. Cotes’
formula for n = 7 gives
(1) f f{x)dx
J a
= ^ {41yi + 2162/2 + 272/3 + my^ + 21y^ + 2162/3 + 41y, }
-6-4(i!>-a)9/<®H5)xlO-i».
Now we have the central difference
(2) S® 2/4 = 2/1 - 62/2 + 1%3 - 20^4+ 152/5 - 6^3 + 2/,.
Between (1) and (2) we can eliminate any pair of functional
values which are equidistant from the central value y^. If, from (1),
we subtract
27 (6 -a)
172 NUMERICAL DIFFERENTIATION AND INTEGRATION [7-23
and thus eliminate y.^, y^, we obtain G. F. Hardy’s formula, namely,
(3) f{x) = -i - ®) { O' 14 (t/i + 2/7) + 0-81 (ya + y®) + MOy^ }
+ 4-6(6-a)7(«)(y X 10-»- G-4 (6-a)9/<'*M5) x lO-w,
since, from 3-0,
mix)-
The coefficients of the remainder term are given to two significant
figures.
If, instead of eliminating one of the values, we add
(6~a) S6 2/4/840
to (1) we obtain Weddle’s Formula, namely,
I /(a:) ix = (6 - a) { (yi + y,) + 5 (y.^ + y^ {y^ + y.J + Gy,j }
-2-6(h-a)V(®>(yx 10-8 -6-4 (6- x
The merit of this formula is the simplicity of the coefficients, the
disadvantage is the complicated form of the remainder term.
The principle here exemplified could be used to obtain an endless
variety of quadrature formulae.
7*3. Quadrature Formulae of the Open Type. If in 712
we put A = 1, we have
h-a
CO = — ,
and consequently
f V(^) dx = {b-a)'Z + V «) +
^ 0>
where
/w- 1^
rnn-x 1
Vv-H
[ >
1 1
0
This leads to the two sets of formulae,
[ fix) dx = {b- a)
dx ■
2m
(6 -a) 2
V =1
V2m + 1/
2w+l
7-3] NUMERICAL DIFFERENTIATION AND INTEGRATION 173
The expressions for G2m-i,v ^2m,i given in sections 7*13,
7-14. In the formulae the functional values f(a),f{b) correspond
to v = 0, v = w + l, and these values are not used in the formulae.
The coefficients satisfy the relation
Z7(«) _ TZ(n)
Xx y
which can be proved in the same way as for Cotes’ coefficients.
The following table gives the coefficients and remainder terms of
some of these formulae :
V
1
2
3
Remainder
2
i
2-8(6 10-2
3
— 1
jj
3-l(6-a)5/W(^)xlO-«
4
1 1
":i‘T
1
•J 4 1
1
VTf
2-2(6-a)5/^«(5)xlO-^
5
1 1
— 1 4^
2 0"
2 6 i
2 0" 1
M(6-a)Vf®U5)xlO-«
6
6 11
‘1'4T1F
4 5 -i
T1T(T
5 6 2
1 4 4F
7-4(6-a)V<®UQxlO-’
7*31. The Method of Gauss. From 7-11, we have
(1) fi^) = S /(®^)
where ^ (x) = (a; - cc^) (<c - ajg) . . . (a; - Xn).
If in the above formula we neglect the remainder term, the
approximation obtained is equivalent to the approximation obtained
by replacing f{x) by an interpolation polynomial of degree n - 1,
which coincides with f(x) at the points x^^, x^, ••• , x^. Gauss has
shewn that by a proper choice of the interpolation points
X2, . • • , Xn
we can obtain an approximation to the given integral equivalent to
the approximation obtained by replacing f{x) by a polynomial of
174 NUMERICAL DIFFERENTIATION AND INTEGRATION [7-31
degree 2w-l. This means that if the n interpolation points be
properly chosen, the remainder in (1) will vanish when f{x) is a
polynomial of degree 2/^ - 1 at most.
Let P{x) denote the polynomial of degree 2n- 1 which coincides
with/(ir) at the points Xq, , Xn+i, Xn+2^ and let
Q{^) = ±
x-x, ^ (x,)^
Q(x) is thus a polynomial of degree n- 1 which coincides with /(a;)
at x^j x^j , 3/^.
Let 0 be a constant. Then P(x)-Q{x) and c<l>{x) both vanish
when x = Xp , Xn, and therefore we have
(2) P{x)-Q(x) = c^{x)N{x),
where N{x) is a polynomial of degree n-1. Then, as in 7*11, we
have
f f{x)dx= f P{x)dx+R,
J a J a
where
(3) jB = I (x-Xi){x-X2) ... {x-Xon) {xx^x^ ... XgJ dx,
J a
Using (2) we therefore have
(4) f (ia; = f Q{x)dx-\-{ C(f>{x)N {x)dx-\-R,
J a J a J a
We now prove that by proper choice of x^, ,x^ the second
integral on the right will vanish.
Let the polynomial resulting from h successive indefinite integra-
tions of <j>{x) be denoted by «^7c(cc).
Then, by repeated integration by parts, we have
[ C(l){x)N{x)dx
J a
since iV<”>(a:) = 0. The integrated expression will vanish if we take
for e^{x) a. polynomial such that
M<^) = 0, <t>„{b) = 0, k = l,2,3,...,n.
7-31] NUMERICAL DIFFERENTIATION AND INTEGRATION 175
This result is therefore attained if we take
fin
[{*-«)"(*- ^)”]-
Since <^(a;) = {x-Xi){x-x^) ... {x-x„), we have
(5) {x - Xy) {x-x^)...{x- x„) = ^ - «)" - ^)"]
SO that the required interpolation points are determined as the roots
of the equation
(6) ^„[(a;-a)”(^-&)"] = 0.
That the roots are all real and lie between a and b is seen at once
by successive applications of Rollers Theorem beginning with
(x-a)^{x-b)^, which has n zeros at a and n at b.
The divided difference in the remainder term (3) is zero if f{x) be
a polynomial of degree 2n - 1 at most, and we have therefore proved
Gauss’ result. It should be noted that when x-^, ... ^x^ have been
determined by (5) the remaining interpolation points ... , x^^
remain arbitrary. If we take 5=1, 2,..., n, the re-
mainder term becomes
jR = f {x- X^^ [x - . . . (x - ^2^2 • • • ^n^n] dx
J a
by use of the mean value theorem.
Since (x-Xj)^{x-X2)^ (x-Xn)^ =: ((l>(x)f, integral in the
remainder term after n integrations by parts becomes
(-1)” £ <f>n{x) <^^^'>dx = (x~aY{x-hYdx.
Integrating by parts n more times this becomes
(-l)2n(^i)3 p {x-aY^ j _ (w!)^(6-a)2«+^
(SH) Ja in+l)(n + 2) ..:^^ [ {2^!]2^+l)'
Thus jSbaally we have the formula of Gauss, namely,
f{x)dx= 2 5'i”V(*s) +
*-i
[(2n)!]»(2M+l)
176 NUMERICAL DIFFERENTIATION AND INTEGRATION [7*31
where ^ lies in the interval (a, b) and iCj, Xg, are the zeros
of (6), while, by 7-11(4),
(n) ^ dx
The advantage of this formula lies in the fact that by the use of
n points only we are attaining the accuracy which would ordinarily
result from the use of 2n points. The disadvantages lie in the fact
that the interpolation points in general correspond to irrational
numbers and their use leads to excessive labour in numerical calcu-
lation.
If we make the change of variable
X =z ^(b-a)t + ^{b + a)^
the new interpolation points t^, ... , tn are given by
n\ c?”
{2n) ! dt^
n\
1.3.5...(2?^-~l)
where P^(^) is Legendre’s polynomial * of degree n, and we have
where the coefficients are independent of the particular interval
(a, b).
The zeros of P„(^) can be arranged in the order ... in such
a way that
s+l = 0,
and if n be odd the middle member of the sequence is zero. -With the
aid of this property it is easy to prove that
= e^.+x-
The following list gives the jSxst six Legendre polynomials :
PM = i(35a;4^30ir2+3), P^[x) = ^a?(63cc^-70a;H15).
* E. W. Hobson, Theory of Spherical and MUpsoidal Harmonics, (1931),
p. 18. See also pp, 76-81, for a discussion of Gauss’ formula and for numerical
data.
7*31] NUMERICAL DIFFERENTIATION AND INTEGRATION 177
With the aid of these expressions the zeros and the coefficients
can be calculated. For the numerical values of the zeros and
coefficients to 16 decimal places for n = 1, 2, , 7, the reader is
referred to Hobson (Joe. cit.),
7*33. The Method of Tschebyscheff.* Let F {x) be a given
function, and (f> (x) an arbitrary function which is assumed to have
differential coefficients up to and including the (w+l)th. We seek
to determine points • • • > such that
where k and the points are independent of the particu-
lar function (/> {x) and where the remainder term depends upon
^(n+i)(2j) only.
We have, by Maclaurin’s Theorem,
/y>2 rpn ,y»W4-l
^x) = <i> (0) + X (0) + ^ 0" (0) + . . . + ^<'» (0) + il),
where 0 < ^ < a;. Consider
l*+i
= J F(x)(li{x)dx-k[<l>{x^)+... + <l>{x„)].
If we put
f+l ^5
T,= \_^^^F{x)dx,
we have
R„ = !,<!> (0) + Tj (0) + . .. f (0)
+ F {X) il) dx-nk^ (0)
- ^i'(O) [a;, + ajg + . . . + a;„] - [x^ + + . . . + -
- [x^+xt + .-+xl]
- + ... + <« ],
where is a number in the interval (0, x^), 5 = 1, 2, , n.
' P. Tschebyschefi, Journal de Math. (2), 19 (1874).
178 mJMBBICAL DIFFERENTIATION AND INTEGRATION [7.33
The terms containing ^(0), ^'(0), ... , ^(«)(0) will therefore dis-
appear if we take
(2) nh=^''^ F{x)dx,
r+i
/c[a;ji-l-a:2+...H-a;J = J xF{x)dx,
^ [®i + ®2 + ••• + = J x''F{x)dx.
The ra-l-1 numbers h, x^, x^,...,x„ having been determined in
this way, (1) constitutes Tschebyscheff’s formula, the remainder
term being
= J-,
wMch vanishes when <j> (x) is a polynomial of degree n at most. In
this case Tschebyscheff’s formula is exact, that is to say, there is
no remainder term.
To determine X23 . . . , x^j we proceed as follows :
Put f(z) = (z-Xj}(z-X2) ... (z-x„), so that x^, x^, ... , x„ are
the roots of the equation
(^) f(z) = 0.
Taking <j>{x) = {z-x)-'^, we have
{x) = {n + \)\(z-. x)-n-2
SO that
where cq, C2, ... are independent of z.
Also, by taking the logarithmic derivate of/(2), we have
L.^m
~ f ii) ’
2-% Z-X^ ' z-x.
and thus from (1) and (6), we have
Cl . c,
-1 z-x
i:
dx=h S:- + -S- 4.
f{z) 2!»+2^2n+3+’”
7-33] NUMERICAL DIFFERENTIATION AND INTEGRATION 179
Integrating with respect to z, this gives
I" iTWlog (»-«)& = ilog .
where 0 is a constant. Taking the exponential of both sides, we have
/(s:) exp (n-h 1) (nH-2)fc2;^+^ “*]
= C exp I F{x) log {z - x) dx^ .
Since the expansion of cap [- ...]
differs from unity by powers of 2; lower than 2:-^, and since /(s:) is,
by definition, of degree n, it follows that the polynomial part of
the first member is equal to f(z), and therefore
(6) /(z) =p|Cexp[^£^J’(a;)log(z-a;)cZa;] |,
where P denotes the polynomial part of the expression in curled
brackets when expanded in descending powers of z.
Since the coefiicient of z'^ in f{z) is unity, the constant C is deter-
mined so that the coefficient of z'^ in the right-hand member of (6)
shall be unity.
By giving particular values to F(x) we can obtain a variety of
quadrature formulae. The most important case is P(a;) = 1, which
2
gives, from (2), A = - . Also, integrating by parts,"
f+i
J log{z-x)dx= (2:H-l)log (2;-fl)-(2-l)log (2;- 1)~2
= 21ogz+(z+l)log(l + ^) + {l-z)log (^l-^j-2
01 2 2 2
-ziogz 2 3^2 4,524 e.Tz® •••’
using the logarithmic series. Thus, from (6),
/(z) = P |cexp[nlogz-2-^- }
" P { ^” [ “ OP ~ O? “ “ ■ • • ] } ’
where we have taken 0 = 1 in order to make the coeflicient of unity.
180 ll^XJMERICAL DIFFERENTIATION AND INTEGRATION [7*33
Taking = 2, 3, 4, 5, we obtain the polynomials
The polynomials are evidently even and odd alternately.
Solving the corresponding equations, we obtain the positions of
the ordinates as follows :
71=2, -x^ = x^ = 0*57735027,
71= S, -x^ = Xq = 0*70710678, x^ = 0.
71=4:, ^x^ = x^ = 0*79465447,
-^2 = 0:3 = 0*18759247.
n = 5, -Xi = x^ = 0*83249749,
-X2 = x^ = 0*37454141, x^ = 0.
Tschebyscheff’s formulae, like those of Gauss, have the dis-
advantage that the positions of the ordinates correspond to irra-
tional numbers. They have the practical advantage of simplicity.
Moreover, when the ordinates are obtained from observation or
measurement and are therefore subject to error, the method has the
advantage that all the errors axe equally weighted.
7*4. Quadrature Formulae Involving Differences.
From the interpolation formula with divided differences, we have
(l) f(^) =f(^l)+ S {x-Xi){x-x^) ... {x-x,) [XjX^ ...
■^{x-Xj){x-x^ ... {x- x„) [xxj^x^ x^].
Let a and a+w be numbers such that in the interval
a<x < a+(i>
the product {x -x^{x-x^ ... (x— x„) has no zeros. In this interval
the product has a constant sign, and hence by the mean value theorem
f«+«
{x-x^) ... (x- a:„) [xxi ... a;„] dx
- f‘‘+“
~ n\ }„
(sC-Kj.) ... {X-Xn)dx,
7-4j NUMEBIOAL DIFEEBENTIATION AND INTEGBATION 181
where lies in the interval hounded by the greatest and least of
o, a+w, Xj., 352, , ^n- Thus we have, from (1),
(2) f ^ /(a;) dx = f(Xi) + [x^ x^] + [x^ x^ x^]
+ ...+A„_i[xiX^...x„] + A„f^ (y/n!,
where
(3) Ag — ~ 1 {x — iCj) (x — x,^ ... {x — cCg) dx.
^ Ja
From this result a variety * of quadrature formulae can be
deduced by assigning suitable values to x^, ajg, . . . , x^.
7-41 . Laplace's Formula. In (2) of the last section put
Xj^ = a, Xs = a + (5-l)(o.
Then
X ra+w
As=~ {x-a){x-a-o^) ...(x-a-s<ji + (jy)dx
Ja
= coM t(t-l)
Jo
Thus, by 64 (4), we have
Also by 3-0,
[x^x^ . . . as^+i] = /I® f(a)ls\.
Thus, since = I*, we have
(1) f(x)dx^l{f{a) + f(a + ci)} + ^Bf'>{l)/i^f{a)+...
Now
r- 1
^ zl®/(a+vw) = J*-V («+»■“)
2
v^l
* H. P. Nielsen, Arhiv.fdr Mat Ast och Fys. 4 (1908), No. 21.
182 NUMERICAL DIFFERENTIATION AND INTEGRATION [7-41
Hence, if in (1) we replace a in succession by
a+6), a + 26), a+(r-l)co
and then add the results, we obtain Laplace’s formula, namely,
(2) Y, ^
CO Ja
^ f (^) (ct + <o) +/ (<X 4- 20)) + ...+• f {t — 1) 0>) + J / (<x +• T(xi) ]
«=2 •
This formula gives the definite integral of a given function in
terms of differences. Alternatively, the formula gives the sum
y(a)+ jr(<x 4- 0)) -f- . . . 4* y'(fl-4'ro))
if the integral can be evaluated. The differences employed are the
forward differences of /(a),/(a4rco).
To calculate these we require the functional values /(a4vo))
from V = 0 to v = n4r-2, and therefore ^ lies in the interval
{a, a4(n4r~2) co).
To find the coefficients we have, from 643,
- v! log(l4^)*
Thus we have
V
2
3
4
5
6
B<;\i)i^\
1 — 1
TT
1
— 1 «
T'jfTr
— H 3
¥ 0 '4 8 O'
That the signs alternate follows from 64 (4).
A corresponding formula for backward differences is easily
obtained by taking = a - (5 - 1) co. We then obtain
ra+w
^i = -J (a:-®)... (a:-a+(s-l)(o)dIa:
= f («+s- 1) ...
Jo
By the complementary argument theorem, = ( - 1)®
7-41] NUMERICAL DIFFERENTIATION AND INTEGRATION 183
Thus 7-4 (2) gives
+ ( - 1)"-^ s^-l” zi"-V(« - (« - 1) <0) + CO" BSrV(">(?).
which is an integration formula with ascending differences.
In particular, for = 4, we have
(3) M ^ /(*) dx = f{a) + i A f {a- oi) + /v Zl V(« “ 2co)
CO Ja
+ IZlV(«-3<o) + fMcoV^^>(5).
Formulae of this type can of course be obtained by direct inte-
gration of the appropriate interpolation formula.
7’42. Formula of Laplace Applied to Differential Equa-
tions. Laplace’s formula with ascending differences is the basis
of the Adams-Bashforth method of integrating numerically a given
system of ordinary differential equations. Such a system can
always be reduced to the first order by introduction of new variables.
Consider the single equation
From this equation by successive differentiation we can obtain
, in terms of x, y. Let it be required to find the solution
with the initial conditions y —y^, x^.
We first calculate y^, j/q", y^'", ... , and then, by Taylor’s Theorem,
y = yo + i^- ^o) Vo + ^ - ^o)^2/o" + • • • •
Taking an interval co for x, we calculate from this series y^, y2, y^
corresponding to ccq+co, cc,)-{-2co, aio+Bco, and from the given differ-
ential equation the corresponding derivates
yx=F{x^,yA, y2=F{x^,yi), 2/3' =
where
Xi = Xq + (£>, == -1- 3o>.
184 NUMEKICAL DIFFERENTIATION AND INTEGRATION [7-42
We can now form the table :
^0
ysl<^
Vo'
AVs
iCo-Hco
2/1/“
Vi
A^yz
Ay{
A^yo
1^0 + 2co
ysh
Vs
Ay^:
A-y,'
Xq “h 3co
ys!^
Vz
Xq 4" 46i
If we can find / oo, the table can be extended another line. Now
by (3) of the last section, if we pnt/(a;) = y\ a = we have
^4 - Jr 2/3 = i/s A Vs + A A'^yi+ il Zl* Vo + f S 0
CO 0)
where is a value of If co be sufficiently small to allow
the error term to be neglected, we have the value of / co, and hence
we can find y'^ = 2/4) = i^(^’o + 4co, 2/4). We can then write in
a new line of differences and proceed to the value of ^5 / 00 by the
same method. This is the Adams-Bashforth process. The extension
to systems of equations leads to greater complexity in the calcula-
tions, but the principle is the same. For an account of the present
state of this subject the reader is referred to a lecture by H. Levy *
on the numerical study of differential equations.
7*43. Central Difference Formulae In 7*4, take
n = 2m, = a + 5co, a:;2,s+i = ^
Then, writing x — a
A2, = {t+s-l){t+s-2) ... {l-s)dt = w** Bg*' (s),
•^0
r (« + S) ... (i - s) (5 + 1).
•^0
* H. Levy, Journal London Math. 80c. 7 (1932), p. 305.
7-43] NUMERICAL DIFFERENTIATION AND INTEGRATION 185
From 6*3 (1), putting x =z s, 1 = v = 2^+1, we have
-S 2^/ (^ + 1 ) = -1(25 + 1 ) J3 (5) .
Also
-2s+l
lx,x^ . . . x^J = (2^31)1 f{a-(s-l) o>),
^-2s
[XjJTo ... = (2^
Thus
2 ta + to m-l / \ f
- J ^ fix) dx = 2 { 4'* /(a - SCO) + 1 /(a - sco)
Now
Zl®’ /(a-sco) + ^J2.+i f(^a-S(i,)
= \A^ f [a- SCO) + 1 Zl^* /(c* - (s - 1) CO.)
= i Zl^-^ /(a - (5 - 2) co) - ^ ^2.-1 y(« _
= !.S2»-i/(a + co)-(xS2-V(«).
in the notation of central differences.
Thus we have
^ = i I /(a) +/(ci + co)|
m~l d(2«)/ \ ^ >.
+ S /(a + co) - (AS2*-i/(a) j
If we write in turn for a the values a + o), a + 2o), ... , a + (r- 1) oa
and then add the results, we get the central difference formula
J ra+rw
- J /(as) dx = {\f{a) +/(a + co) + . . . +/(a + (r - 1) «) + i/(a+ rco)}
~fh)f + »•«) - !J'S^'"V(a)}
186 NUMERICAL DIFFERENTIATION AND INTEGRATION [7.43
The differences actually used with this formula are shewn
schematically as follows :
S
2/o=/(«)
Vt =/(a + ra)
Thus, for m = 3, we have
T
6-aJa ^iyo + yi + y--+ ■■■ + yr~l + iyr]
- J (%r-i + Syr+i) - 1 (Sy-i + Syj) }
11
720 { i (S®yr-i + - i (S®y-J + }
191
W480 '
Example. Calculate 1 sin xdx,
J *160
using the table
X
sin a;
A
A^
A^
9873068
-10
0-160
0-15931 82066
9871475
-1593
-11
•161
•16030 53541
9869871
-1604
-8
•162
•16129 23412
9868259
-1612
-11
•163
•16227 91671
9866636
-1623
-10
•164
•16326 68307
9866003
-1633
-9
7-43] NUMERICAL DIFFERENTIATION AND INTEGRATION 187
Thus, since r = 4, b-a = 0-004, we have
r'i64
1000 sin xdx:= 0*64516888105 + 0*00000005377
J-160
+ *0032x4x (-001)6 X *16.
The remainder term, affects only the 21st decimal place on the
right. We therefore obtain
f-164
sinxdx = 0-00064 51689 (348),
J-160
the figures in brackets being actually given by the above numbers.
The correct value to 15 places, namely,
cos -160-008-164 is 0-00064 51689 34801,
so that the above result is correct to 13 places. The precision of
this result may be contrasted with the loss of accuracy in differ-
entiating a table, as in the example of 7-02.
7*5, The Euler-Maclaurin Formula. Denote by Py(x)
the periodic fimction of period unity which coincides with By{x)
in the interval 0 ^ a; < 1, so that
P^(x) = B^{x), 0 < a? < 1,
Py{X^l) = P,(x).
Since jB^(1) = B^{0) if v > 1, we have
P,(1) = P.(0) = 5,(0) = 5.(1),
so that P, (x) is a continuous function at x =1 and therefore at
a? = 0, 1, 2, ... , provided that v > 1. Pi{x) is, however, discon-
tinuous at these points, for Bi{x) = cc- -I, and hence
Pj(a;) = a;-|, 0:^a;<l,
P,( + 0) = -i Pi(-0) = +|.
Again, from 6-5 (5),
DB^ix) = v5,_i(a?).
Thus D Py (x) = V P,_i (a?)
and therefore D Py (x) is continuous at 0, 1, 2, , when v = 3, 4, 5, . . . ,
but DP2(cc) = 2Pi(cc) is discontinuous at these points.
188 NUMERICAL DIFFERENTIATION AND INTEGRATION [7.5
Now consider the expression
(1)
E..=
CO
m
Integrating by parts, we get
=
m!
C^m-l rl
~ J 0 - 0 /<”•-’) + <0i) di.
Since P„(y- 1) = P^(y) = B^{y), we have
(j^ni
^”^ = - m\ ^ (*) + .
^2 2 1 ^2 (y) A (^) 4" Ri<
Now
I2i = -.<af Pi{y-i)f(x+vit)dt
J 0
= -« f {y~t-\) f{x+(Sit) dt-<i> f {y-t+D f'{x+ii>t)dt
J y
■ =/(^ + «/w)--Bi(y)<oA/(a:)-- r '“/(«)
to CO Jjj;
the last line bein^g obtained by integration by parts and then writing
t for a;+col in the integral. ®
Thus, by addition of all the above equations, we have
to X/ . ^
-%P3(y)Af''{x)-
* a>
co^
ml
Pr.(y)Af”‘-^K^)~E„.
U>
is the general Euler-Maclaurin formula, of which 6-511 (3)
particular cases when./(a;) is a polynomial
7-5] NUMERICAL DIFFERENTIATION AND INTEGRATION 189
If we put 2/ = 0 and write 2m for m, we have,, since
= — i, = 0, 5 > 0,
1 ra: + " w ,.2s
w J a; “ i { /(^) +/(^ + o)) } - A (a;)
(2^) I J Q (0 {x + Oit) dt,
since
P2m{-i) = - ^2.(1-^) = B,^{t) = P,^(0.
If in (3) we write in turn x + co, a;+2o), ... , a;+ (?i~ l)co for x
and add the results we obtain
1 fx+nta
wj* {^/W + /(»+<:^) + /(a:+2co) + ...
+ /(a:+(M- 1) co) + |-/(a:4-nco) }
m-l q\2s-1
- 2 ^ { /<2’-1>(2- + Ww) } + S2„,
where
^2m-l
S2m = - -(2^^y| {/(2«-i) (x + nco) -/(^^-D (;c) }
0)2’«
■b (2m) ! J Q /(2m) ^2; 4- (0^ + SO)) dt.
Now (^ + 5) = Pg^ (0, and therefore
S f ^2m(l)/^^”‘>(^ + 0ii + SC0)d( = r P,Jt)/(^-^)(x + (Ot) dt.
s~0 *'0 Jo
Thus we have
^27n fn
= (2m)! Jo (^2j()-B,J/<^”‘>(x+<ot)dt.
Now
[ (P2m (0 ” -®2m)
Jo
n-1 ^^+l ri
= S (A™ (<) - B,J dt = n\ (P,Jt) - B,J dt,
since PzmW = Am(*+®)> ajid P2mW = P2m(0 tlie interval
0^i<l, so that the last integral is equal to -nSgwt* Also,
190 NUMEBICAL DirEEBENTIATION AND INTEGEATION p.g
by 6-63, does not change sign. Hence by the mean
value theorem
(Vi K
o<e<i.
_ Thus if /(2™+2)(«) have the same constant sign in the
interval x<t <x+'ym, have opposite signs (since
•®2m> Pim+i liave opposite signs). In this case it follows, as in
3-12 (c), that the error in (4) due to neglect of the remainder tennis
less numerically than the first term omitted from the series (4) and
is of the same sign. Again, if /(2«) {t) have a fixed sign in the interval
x<t<x+im,v& have, from (5), by the mean value theorem
^2?n-l
~ pm)] ^ {x+im) -/(2m-i) {x))M,
where M denotes a mean value of P2„{t) - B,
Now, by 6-52j
2m
Thus I if I ^ I I , so that the error in (4) due to neglect of the
remainder term is in this case numerically less than twice the first
omitted term and has the same sign.
If in (2) we put y = | and proceed as before, we deduce another
formula which is sometimes useful, namely,
I roj+no)
= ^/(*+i“)+/(*=+|-«) + ---+/(aJH-(w-|.)co)}
~ (25)! ® -y(2s-l)(2;) }
As an example of (4) applied to the interval (a, b), we have for
n = B, m = 3, w = (6-fl)/3.
7-51] NUMERICAL DIFFERENTIATION AND INTEGRATION 191
7*51. Application to Finite Summation. With the
notation of 2*6, 2*7j we have from 7*5 (4),
rn w— 1 P
This formula gives the value of
P(n) = Mi + M2+-" + M»-
For example, taking m = 1,
l^(«) (a; + i4-l)2 cc+1 a; + w'^2(a;+ l)^^2(xH-n)^^
Since and have the same constant sign, we have
= o<e<i.
If we let n 00 , we have therefore
,§ (a: + fi)2“a;+l'^2(a:+l)2i + 6(a:+l)3’ 0 < 0 < 1.
By taking x large enough we can make the last term as small as
we please. We can therefore find by this method the value of
% 1 ^1
“2 required degree of accuracy by first calculating 2/ 72
by addition, having chosen x sufficiently large. If we take a larger
value of m the calculation is of course made more expeditious.
7*6. Gregory's Formula. With an obvious notation, 7*5(4)
can be written in the form
1 r^+nw
(1) 7 = i fix) (ia;- (I/0+/1+/2 + ... + /n-i+i/«)
Ja
192 NUMEBICAL DIFFEREN-TIATION AND INTEGRATION [7*6
If we use Markoff's formulae of 7*02 we can express in
ascending differences and in descending differences. We thus
obtain
-60
2w~3 p(»'+l) /i\
=^-i{v-2s+l)!
~ l)*’/n-^ + /o }
,v2m~2
(2w-2s-l)!
BfZ-iU (1) ,
Thus we have
I-R--
where
(2) R = -
m— 1 R 2m— 3 t>(i'+1) /-in
X'' jr?l ^y-2s+l{^) y1y((_Uy f +f\
h (2^)! .=^^-1 F-25+1)! ^ + J >
A/) f \iu7Tlf
'm-~l P p(2?w--l) /-j\
+ V ■^2.s^2m-28-lll; 2r.i2m~2
Vi (25)! (2m-26-~l)!
where 0 < 0 < 1 and lies in the interval {a, a + noo).
Hence we can write
2?n~3
I-R= g KAA''fn-y+i-iyA’'fo}>
where
V r; n, - 2!(v-l)! 4! (v~3)! 6!(v~5)l
the series ceasing when the suffix becomes negative.
If in (1) we put f(x) = Bi''^i\x+l)/{v-i~l)l, a = 0, n = l,
the remainder term will be zero if m be chosen large enough, since
f{x) is a polynomial. Also, by 64 (1), /(I) =/(2) = 0.
We have, therefore,
r ^y+i^i^+- 1) Jr - - V Bi’'Ai>+.A2)-Bi-'Ai>+., (1)
)o (v+1)! .tl(2s)! (v-25 + 2)!
Thus
._V J?i
;i'i(2s)! (v-2s + l)!
and we see from 741 that the are all negative but numerically
equal to the coefl&cients in Laplace’s formula.
1-6] NUMEEICAL DIB’FEBENTIATION AND INTEGBATION 193
We thus obtain Gregory’s formula, namely,
1
- (i/o +/l +/« + • • • +/«-! + i /n) - xV ( ^ fn-1 - A fa)
CO Ja
- * iA^fn-2 + Zl Vo) - rVVy Vn-3 “ Zj Vo) " itu (zl Vn-4 + A* fa)
-■■■ + 2) ! - A^^-^fa)+R,
where R is given by (2).
The advantage of Gregory's formula, as compared with that of
Laplace or the central difference formula, lies in the fact that the
differences employed are the descending differences of /(a) and the
ascending differences of f{a + n(f>), all of which can be formed from
values of the function within the range of integration. The dis-
advantage lies in the somewhat complicated form of the remainder
term.
7*7. The Summation Formula of Lubbock. Suppose
are given a
table of a
function in the form
n
f{n)
0
/(O)
zl/(0)
0)
/(I)
zlV(O)
zJ/(l)
2co
/(2)
•
(r- 1) CO
fir-1)
Afir-1)
rco
fir)
Afir)
A^fir)
and that we wish to subdivide the interval into h equal parts and
then form the sum
/(0)+/®+/(D+-+/(?)-
194 NUMERICAL DIFFERENTIATION AND INTEGRATION [7.7
The summation can be effected in the following manner by a formula
due to Lubbock.
We have, by Newton’s Interpolation formula,
/(w+l) =/(«) + (^(^) + A^f{n)
Summing from 5 = 0 to A - 1, we get
^^ /(«'+!) = V(«') + -Y- ‘^f{'>^) + \A^f{n)
If we now sum both sides from w = 0 to n r-1, we get
/m\ f* — ! "L ir-l
So ^ So ^
+ ^2 2 A~f(.n) + >^3 2 Zl V(^) + »■ ^4 (^2)
= A 2 /(V) (/(r) -/(O) ) + X2 ( J /(r) - A fiO))
+ MA^ f(r) -A^ /{(>)) + r\u>^ /W (^2) .
Adding to each side, we have
So^il) = ^ ) + ^2(zl / W - J /(O) )
+ >^( f{r) -A^f{0)) + r\ /w (y ,
wHch expresses the required sum in terms of the given difference
table. ^
The above formula has been arranged to include second differences
Clearly any required number of differences may be included bj
7‘7] NUMEBICAL DIFFEBENTIATION AND INTEGBATION 195
taking further terms in the original interpolation formula. To
calculate Xj, X3, ... , etc., we observe that
Ti+ + 2 — =-.
^ ,'^0 (l + a;)i''*-l
Thus
* = (* + i(^-l)*+X2x2 + ...)((l + a:)i/ft-l),
whence
so that the coefficients \ may be successively evaluated.
We have in this way
X3 = -
h?-l
12A ^
X3 =
. _ (P-1)(19A2-1)
24A ’ ^ 720P
Example. Calculate, by Lubbock’s formula,
3530 1
I
3^ ^
Taking = 10, r = 3, we form the table :
n
3500
0-0002857143*
3510
2849003*
3520
2840909*
3530
2832861*
3540
2824859
3560
2816901
+
8140*
8094
8048
8002*
7958
46*
46
46
44*
The munbers used in the formula are indicated by an asterisk.
196 NUMERICAL DIPEBRBNTIATION AND INTEGRATION [ex. vn
The required sum multiplied by 1(T-® is equal to
10 X 11379916 - 4-5 x 5690004 - x 1,38 -
where
P 10“x 3x10^x99x1899x24
li< — 1 . rrctri . . /ocr ArwR *UU4:j
103 X 720 X (3500)^
which is negligible. Thus the required sum is 0*0088194027.
EXAMPLES VII
1. Taking 10 figure logarithms to base 10 from x = 300 to
£c = 310 by unit increments, calculate the first three derivates of
logcc when x = 300, x = 305, x = 310.
2. From a table of sin x verify, to the number of figures which
the tables permit, that
d .
^ sin cr = a cos x,
where u = 1 if a; be measured in radians and a = 7c / 1 80 if cc be
measured in degrees.
3. By means of tables, calculate
Jsiaxdx, sinxdx,
n
using an approximate integration formula.
L The two radii which form a diameter of a circle are bisected,
and perpendicular ordinates are raised at the points of bisection.
Required the area of that portion of the circle, included between the
two ordinates, the diameter, and the curve, the radius being supposed
equal to unity. Compare the result found by Weddle’s rule with the
exact result.
5. Prove that
™ TT
1 logsinOde = - r OcotOdO,
-Jo Jo
and hence calculate the value of the integral by Weddle’s rule and
estimate the error.
EX. VII] NUMERICAL DIFEERENTIATION AND INTEGRATION 197
6. Shew that Simpson’s rule is tantamount to considering the
curve between two consecutive odd ordinates as parabolic. Also,
if we assume that the curve between each ordinate is parabolic and
that the curve passes through the extremity of the next ordinate
(the axes of the parabolae being in all cases parallel to the axis of jr),
the area will be given by
^ [ s 2/ - 2 V { 15 (yo + 2/n) - 4 (^1 + + (^2 + y„_2) } ].
[Boole.]
7. Prove that approximately
8 fSpA
^ ydx = yo+^(3ji+yi+yi+y5+yi+y&+ ■■■)+yzp
+2(y3+-y6+y9+ •••+y33>-3)>
and find the error term. [Newton.]
8. Prove that approximately
45
2h Jo ^ 'lyo + 14(y4-l-^8 + ...+y4j,_4) + 7^4P
+ 32 (j/i +^3 +2/5 + . . . + y43,_i)
+ l2(y2+y6+yio+"-+y4»-2)>
and find an expression for the remainder term. [Boole.]
9. Prove that approximately
rnh
ydx = Ti{y^ + yi+...+y^_^)-lli {y^ -yo + y„^ - y„) ,
and find the error term. [Poncelet.]
10. Prove that approximately
j*nh
ydx = h{yi+y.i+...+ y„_j) - (y* - yo + y«-} - y™) ,
Jo
and find the error term. [Parmentier.]
11. If f{x) be a polynomial of degree 2n~l at most, prove that
^ f+l f(x) f TZ\ , rf 3tc\ . . (2^^-l)7C^
[Bronwin,]
198 NUMERICAL DIFFEEENTIATION AND INTEGRATION [bx.vh
12. Find an expression for the sum to n terms of
,111
l + p + 92+iP+->
and calculate approximately the sum to infinity.
13. Find the sum to infinity of
1 i i ^
correct fo 10 decimal places.
14. Find approximately the value of
p-i 1
(jF+1)2^^2
and obtain an exact formula when a is an integral multiple of
15. Shew that the sum of all the integral negative powers of all
the positive integers (unity being excluded in both cases) is unity ;
if odd powers be excluded the sum is |.
16. Prove Burnside’s * formula for double integration :
+tV{/(^) 6)+/(^, - b)+f{-b,b)+f(-h, -6)},
where o = Vtt) ^ = and/(a;,y) is a polynomial of degree 5
at most.
17. By successive applications of Simpson’s rule obtain the
formula for double integration :
+f{2b, 2c) +f{2b, 2d) + 4 [/(2a, c + d) +f{2b, c + d)
+f{a+b,2c)+f{a+b,2d)] + 16f{a + b,c+d)},
and investigate the form of the remainder term.
* W. Burnside, Mess, of Math., (2), 37 (1908).
EX. vn] NUMERICAL DIFFERENTIATION AND INTEGRATION 199
18. Prove that: (a) By elimination of the error term in
between two trapezoidal formulae corresponding to sub-intervals h
and 2A, the Simpson Rule is obtained, {b) By a similar elimination
between formulae corresponding to h and Cotes’ formula for
=4 (the Three-Eighths Rule, Ex. 7 above) is obtained, (c) By
elimination of the error term in between the Simpson and the
Three-Eighths rules for 7 ordinates, Weddle's Rule is obtained.
[Sheppard.]
19. By eliminating the error term in between two Simpson
formulae of 2_p-bl and 4p-fl ordinates, obtain the Newton-Cotes’
formula for ^^ = 5.
20. Obtain the following table of quadrature formulae of open
type, in which the ordinates used are at the midpoints of equal
sub-divisions of the range {e,g. for 4 points and range 0 to 1, the
ordinates used would be f, |-) :
V
n
1
2
3
Remainder
1
! 1
Mh-aWm
2
1
i
i
1
3
¥
0
t
1
4
1 3
1 1
4 "8'
1 1
4 8
7-0(6-a)6/W(^)xl0-s
5
2 7 i5
lT-§-2
10 0
TTS-^
4 0 2
TTT^r
3-7(6-a)V'®H?)xlO-’
The remainders in all cases have positive sign. [Aitken.]
21. Numerical differentiation of a tabulated function y{x). Let
y'(^x)=bQ+bjX~{- . . . 'j-bnX”'. Minimise
[6o+ . . . +bnx^—y'f
dx==Y, giving dV jdb,
dbs=0,OT J
(60+ • • • +bnX?^) ^dx=
y'afdx=^
| x^'~'^y dx=K8, whence we get w+1 equations to
determine bn in terms of kq, /c^, .. .3 Kn which can be calcu-
lated from the tabulated values.
CHAPTER VIII
THE SUMMATION PROBLEM
CoNSiDEE the difference equation
Au{x)==<l>{x),
0}
where <!> {od) is a given function. The sumnaation problem consists in
determining u{x). If u*{x) be a particular function which satisfies
the equation, and if w{x) be an arbitrary periodic function of x of
period 6), it is evident that (x) + w (x) is also a solution. That this
solution is the most general possible is seen from the remark that
the difference of any two solutions is a solution of A'^(^) = 0, so
to
that any two solutions differ by an arbitrary periodic function of
period co. Thus if ^(a;) = 2x, the most general solution of the given
equation is x^-o>x+wf(x), particular solutions being
x^ ~ cocc, x^-c}X+ sin (2kx / co) , - coa; + 1 co^.
That particular solutions of the given equation always exist is seen
{in the case of the real variable) by considering that u(x) being
arbitrarily defined at every point of the interval 0 a; < co, the
equation defines u(x) for every point exterior to this interval. Such
solutions are in general not analytic. The problem of determining
analytic solutions has been studied by many mathematicians. In
this chapter we shall consider solely Norlund’s theory of the equation.
By an extremely elegant method Norlund has succeeded in defining
a principal solution ” which has specially simple and definite pro-
perties. In particular, when ^(a;) is a polynomial so is the principal
solution. Moreover, the solution is defined by an algorithm which
supplies the means of obtaining the solution. We can only study
200
THE SUMMATION PROBLEM
8-0]
201
here the more important outlines of the theory. For further details
the reader is referred to Norlund’s memoir.*
8'0. Definition of the Principal Solution or Sum. Con-
sider the difference equation
(1) Au(x) =
Ul
or u{x + o^)-u{x) ~
The expression, (where A is constant),
f{x) = A-o^[(f)(x) + <f){x + o^)-hcl){x + 2o^)^(j){x + Z(^) + ...]
CO
8^0
is a formal solution of the difference equation, since
/(oj+co) = [9!>((rH-<ja) + <5i(x + 2co) + <^(x-f3oi)-f ...]
and therefore /(cc-f co) -f(x) = <o
If for A we write J ^ (t) dt, and if this infinite integral and the
infinite series both converge, we define the frinci'pal solution of the
difference equation, or sum of the function (j>(x), as
(2) F{x\o^)={ ^ (l>{x + so^).
The reason for the introduction of the infinite integral will appear
shortly. The principal solution thus defined depends on an arbi-
trary constant c. We may thus consider the principal solution as
being formed by summing ’’ the function (x), and from this point
of view, by analogy with the notation of the integral calculus,
Ndrlund writes
(3) jP(ci5| co) — Q ^ = f S +
O " g^Q
C
and the process may be referred to as '' summing ^ (a?) from c to xF
* N. E. Norlund, “ M^moire sur le calcul aux differences finies,” Acta Math,
44, (1923), pp. 71-211. See also Differenzmrechnungf ch. iii. The examples at
the end of this chapter are all due to Norland.
THE SUMMATION PROBLEM
[8-0
202
As an example, consider
X and CO being real and positive. Here
» r*
(4) ^(ajlco)^ ^ e-*A2 = J_
CO e~
after evaluating the integral, and summing the Geometrical Pro-
gression.
The necessary and sufficient conditions for the existence of the
sum F{x\(j^) as defined above are the convergence of the integral
and of the series.
In general, neither of these conditions is satisfied and the definition
fails. In order to extend the definition of the sum, Norlund adopts
an ingenious and powerful artifice. This consists in replacing (x)
by another fuhction, of x and of a parameter p. ( > 0), say <p{x, p),
which is so chosen that
(i) lim <l>(x, p)
(ii) r <j) (f, p) dt and
00
^ (f){x + s<x, p) both converge.
Por this function p), the difference equation
(5) A'i^i^) -
w
has a principal solution, given by the definition (3),
C30
(b) F{x\(x>, ^ I p) dj/'— CO ^(x+sco, p).
If in this relation we let p 0, the difference equation (5)
becomes the difference equation (1) and the principal solution of
the latter is defined by
F(x 1 co) = lim F{x i co ; p),
provided that this limit exists uniformly and, subject to conditions
(i) and (ii), is independent of the particular choice of p). It is
g.O] THE SUMMATION PROBLEM 203
of course assumed throughout that the domain of variation of x and
CO may be subject to restrictions depending on the nature of the
function {x) . The nature of these restrictions will be more apparent
when we consider particular classes of functions. It may also be
observed that the success of the method of definition just described
depends on the difference of the infinite integral and the infinite
series having a limit when ^ 0. Each separately may diverge
when ^ = 0 and the choice of <f>{x, p.) has to be so made that when
we take the difference of the integral and the series the divergent
part disappears.
When <f){x) is such that the sum exists, it is still possible that the
result obtained may depend upon the particular method of sum-
mation adopted. In this connection it has been shewn * that, for a
wide class of summation methods, the result is independent of the
method adopted. Among these methods is the one given in the
following definition, which will suffice for our purposes.
// for X variable in a certain interval and for positive values
o/co, we can find 1, 0, such that for l{x) = a;3’(log
f dt, 2 ^(a;+5co)
Jc ^0
both converge for p > 0, then the principal solution of the difference
equation A i^
(7) F{x\o,)^^<l>{z)l^z
c
= liini
) V Jc fdo J
provided that this limit exists.
When the limit exists (j){x) is said to be summable.
As a simple illustration, consider
A u(x) = a,
ta>
where a is constant.
* N. E. Norlimd, Acta Univ. Lund. (2), 14, 1918 ; 16, 1919.
THE SUMMATION PROBLEM
[8-0
204
The series a + a + a+ ... obviously diverges, but for (j, > 0
^00 00
ae-i^dt, ^ a e-'‘ (*+*“'>
Jc ^0
both converge if co be positive, so that we can take X(a;) = a;,
i.e, jp = 1, g = 0. Hence
^ A 2^ = lim < 1 a di-cx> a (
W fl — >-0 1. J (
-IX (a:H-^u))
— lim ^ ^ ^ ^
a e~>“([L(x> - <0^ + ... - [xc«> + CO (cc- c) - ...)
.•m ^ ■
lA(^[i.CO-i^+...J
: lim
^—>■0
(8) ^ aA 2= ®(!»-c-4ce),
which is the principal solution. It should be noted that both the
integral and the series diverge when p. = 0. For o = 1, c = 0,
a = 1, we have
%
Q 1Az = ®-| = 5i(®)-
0 “
Thus Bernoulli’s polynomial B^{x) is a principal solution. That
all Bernoulli’s polynomials are principal solutions will be proved
later.
8*1. Properties of the Sum. We shall now consider some
of the general properties of the sum which are directly derivable from
the definition.
If in 8*0(6) we write for x successively the values
m m m
and add the results, we get
“S co; ^) = mF(x y),
8^0 V m \
THE SUMMATIOK PROBLEM
205
which when pi 0 gives
\ m / \ m/
and also
2 m-l
(2) J’(a:lco) = - F{x+sci\mo:>).
nff a=o
These results constitute the Multiplication Theorem of the Sum
(cf. 6*52).
Again, from the definition,
F[x — ;pi)=f +
m Jc w \ m
so that if m -> 00 ,
lim Fix -
771— >00 ^ ^
SO that when [x — > 0,
lim F I X
Again, from (1),
; F-) = | ^{t,\x)dt
V T 1 C) n
lim Fix — ) = lim \ F (x + — co
^ m/ ^^00 CO m s^o \
1 ra;+«
Thus we have the result
ftc+w rx
I F{t\<o)dt = <l>{t)dL
' Ja; Jc
From the definition of the srun it follows that
(5)
206
THE SUMMATION PROBLEM
[8-1
'JC
Tims tke operator A cancels tie operator ^ A On the other
Tifl-nd^ these operators are not commutative, for
(6) §[A#«)]Ai=ig#(i+-)A«-i§#WAi
c
rc+to
OJ (O u>
c+«
X
e
1 fc+w
= 4i^)-^A <l>it)dL
COJc
From this formula we caa derive a rule for summation by parts of
the product of two functions. For writing
we have
^(z) = u(z) Jsl v(t)At,
- 2+«o Z
A9^(z) = -w(2 + oa) Q v(t)At--u(z) Q t!(t)Ai-
„ a O u w u „
Thus
S-t-w S
A^iCz) = [Aw(2)] Q v(t)A(+u(z)A Q v(t)At,
w (0 \_y to to w
C C
so that by using (5),
Z+to
uiz)v{z) = A<^>(z)-iAu{z)] Q v{t)At.
to to to
c
Sum both sides from c to x, using (6), and we have
(7) ^ m(2)«(s) Az = m(*) ^ ®(«) Ai-^ J w(2) ^ «(i) A^<^Z
c c c
X z + w
- § i[AM(z)] ^ 1^WA«}Az.
I, to to J to
THE SUMMATION PROBLEM
20T
8-1]
This formula is quite analogous to the formula of the integral
calculus for integration by parts.
As an illustration, take m(z) = z, v(z) = e-*. Then, by 8-0 (4),
To evaluate
X
we use 8-0 (8), so that after reduction
§,e-A^
<0
+ ce“^ + e"^.
We have thus found the principal solution of the difference
equation
= xe~^.
We also note that when co 0,
lim
«— >0
X
c
ze~^ = -X - er^-\-o er^ +
z dz,
a result which we should expect in view of (S), which points to the
result
lim
*0— >-0
That this is true when <o tends to zero by positive values will be
proved in 8-22.
The following identities are often useful and are easily proved :
(8)
u) J a
dz,
x+a
^ ^(z) A2 =
X
^(z+a)^z.
w
208 the SUMMATIOlSr PROBLEM [8-1 1
In tiiis connection we note, in contrast with the known result
c
cj> (z) dz = 0, that ^ ^ (z) A 2 is not zero in general.
C
8-1 1 . The Sum of a Polynomial. Consider the equation
/^u{x) = nx”'~^ e~i^.
Taking c = 0, the principal solution of this is
^00
F(x\l: u.)= w(a;+s)”-ie-»‘(*+«)
Jo
Thus when (x 0, we have
F{X\1):=B,{X).
We have thus proved that Bernoulli’s polynomials are principal
solutions or that the sum of nx^~^ from 0 to a; is B^ix). It follows
that the sum of a polynomial is a polynomial.
8’12. Repeated Summation. Consider
5 \
Tn / I \ -I ' 1 1 I / .\ A .
F,{x\oy) = oy-~^^ co ^
We have
a;+a> / x-t
jF^(a?+Ci) I co) = ca'
THE SUMMATION PROBLEM
209
8-12]
SO that
J„(a; + co|co)--F„(a:|co)
fX + o /x—t"
a /x-V
An-1
CO )^(^)A^
25 /x—t
-1
W\n-2/
The expression in the curled bracket is zero, for if
4'(0 ~ ^ ^
we have (a?) = 0. Hence
A^’„(a;|w) = -F„-i(a:|6i).
Thus
A^nla^l w) = 4>{x).
We shall call | co) the nth. successive sum of and write
25 /x-t ^
O) I ^ (i) A *•
n-1 .
8-15. Proof of the Existence of the Principal Solution
(Real Variable). We prove that, under certain restrictions on
j>{x), and for positive values of co,
(1) g .^(i)e-^WAi
C
= f°° ^(i!) e-e'^^'^dt-co 2 ^(a:+Aco + sco)e“e^(*+*“+*“5,
J c <=0
where <1, exists and, when p,->0, tends uniformly to a
OJ+Aw
limit ^ A
c
THE SUMMATION PROBLEM
210
The proof
7-5 (2).
[8-15
IS
based on the use of the Euler-Maclaurin formula
To fix ideas we make the following assumptions :
(A) For c, D'^cf>{x) exists and is continuous, where D denotes
the operator ^ , m being a fixed positive integer.
(B) lim D“^(x) = 0.
a;— >00
OO
(C) 2 + is uniformly convergent in the interval
c < X < 0+6), and consequently by (B) in every interval c^x^b
however great b may be.
From (B) it follows that
(2)
lim
== 0, lim
= 0.
With these restrictions on <i>{x)^ the convergence of the integral
and series in (1) is assured if we take l{t) = t, and both sides of (1)
exist.
From the periodic property of Bernoulli’s function (x), we have,
as in 7-5,
rn+3>+l
(3)
Jn
D^<^{x+to:> + so>) dt.
(C), given s > 0, we can choose Uq, so that
nfp
2 P”^</>(a;+^o> + 5co)
< e, for n >
Also P^{-t} is bounded. Hence the left-hand member of (3) can
be made arbitrarily small by choice of n only, and consequently
W f Pmi'^i) dt
JO
is uniformly convergent.
THE SUMMATION PKOBLEM
211
8-1 5]
For brevity, we define operators Q, by the following relations :
(6) Qf{x)=S:'^BAh)D''~^f{x),
v = l ^ *
^.■vW rn
(6) r„ f(x) = -j- P„{h-t)D”‘f{x + co«) dt.
Then, from the Euler-Maclaurin formula 7-5 (2), we have
(f>{x + ha) = - <f>(t)dt + Q{<f>{x + (^)-<l>{x))-Ti<l>{x).
0) J X
Writing in succession x-to, x+2o), , x-h(n~l)cd for x and
adding, we get
n~ 1
(7) 2 ^(ic + ^co + sco)
8 = 0
1 rz+noi
= - dt+ Q <f>(x-{-n<^) - Q (j)(x) - T n4>(x),
^ Jz
Now, by Leibniz’ Theorem,
(8) {- iiY <j>{x).
If therefore in (7) we write (j>[x)e-^^ for cl>{x) and let 7i-^oo,
we have Q(/>(x + nco) and therefore, from (1),
z-\-hoi
(9) g
J C V = l
+ u>T„<l>(x)e-'^=^,
where, by (6) and (8),
(10) <^T^4>{x)e->^^
= { £ Pm {h -t)[D”^<l>{x + (^t)]e- dt
)■(:)]
Jo
We now prove that ^ 0 uniformly, when yL — > 0.
212
THE SUMMATION PROBLEM
[8-15
JL U.U
f{t, (Z.) = [A P^{h-y)e-'^y dy.
This integral converges when [x > 0, since P^{h-y) is bounded.
Then
f{t, (i) = [A f P^{h-y-t)e-i^'‘<^+^dy
Jo
00 ps+1
= tAe-e“‘2 P^{h-y-t)e-i^'“ydy
ri
= [Ae“e<»* 2 «-»*»" I Pn{h-y-t) e-y-“ydy
s=0 Jo
p-u.(at ri
= \PmQi-y-t)e-'^'“vdy.
Hence lim f{t, ti.) = ^ f P^{l-y-t) dy = 0,
since P^{ir) lias period unity. Now, integrating by parts,
ly = /(O, p.) + co r D^-^+^(j>{x + o>t) f{t, p.) dL
J 0
Thus J,, “> 0 when p -> 0.
We have now to consider the first term in (10),
Jo
Put ^ (t) = P„(h-y)P”>(f,(x+ Cdy) dy,
which by (4) is uniformly convergent. Then, integrating by parts,
^ (0) - p oi I ij; (i) dt.
Jo
Thus when p —>0,
Jo
Thus finally, from (9), we have
x+hoi
SCx ^
^(i)A<= <f>{t)dt+'2^ B,{h)D’'-^<f>{x
« J C V=:l V !
c
^m+l Cco
+ — r P.„,Qi~t)I)^<i>{x+(x>t)dt,
ml Jo
g.jgj the summation problem 213
We have thus proved the existence of the principal solution or
sum under the restrictions on ^{x) enumerated above, in particular
for all functions which increase less rapidly than a polynomial of
arbitrary degree.
8*16. Bernoulli’s Polynomials. In 8-15 (11), take
h=0, <f>(x) = va:-'-!, m = V, c = 0.
Then we have
aj V / \
Q V A < = 1 '' + S *’'"'(1)
vy w Jo s — 1
which again shews that Bernoulli’s polynomials are principal solu-
tions, a result already obtained in 8-11 for the case w = 1. If we
define Bernoulli’s polynomial of the first order, of degree v, by the
relation
we Ifave proved that
B,(a;|co) = l),
which gives B, {x \ co) in terms of the polynomials of Chapter VI.
We have quite readily
8t2. Differentiation of the Sum. From 8*15 (11) we
have, on account of the periodicity of W uniform con-
vergence of the infinite series,
(1) §
c
rjp^(k-t) 2 D’”(j>(x+su>+0it)di.
ml Jo s=o
THE SUMMATION PROBLEM
214 THE SUMMATION PROBLEM [8-2
Taking m > 1, differentiate both sides with respect to the para-
meter h and divide by ce. Then by 6-5 (5), we obtain
7 m ,iv-l
+
(m- 1)
iJo S==0
'a; + 5co + co^) (Zif.
Putting A = 0 and writing v for v ~ 1 this can be expressed in
the form
c
/,% W f 1 “
+ jji J ^ w-1 ( ““ ^ + 5CO + (oi) dt.
Comparing the expression on the right with (1) when A = 0, and
m- 1, j>{^) written for m and (/>{x), we have
X iC
(2) §<A'(0Ai+9^(c),
which shews that the differential coefficient of the sum differs from
the sum of the differential coefficient by the constant (f>{c).
8*21. Asymptotic Behaviour of the Sum for Large
Values of a?. With the same hypotheses as in 8-15, we have
from 8-15 (11),
X
^ 9^(0 A« = 6m(a5)+i2„(ir),
a,
C
Cx ^ f
(1) where Q,„(a;) = <f>(t)dt+^
J C v=l Vi
^m+l Too
We have seen, 8'15 (4), that the integral is uniformly convergent.
Changing the variable, we have
Bm{^)
^ f" p (^-y)
m! Ja, “v CO ) dy”'
^{y)dy.
THE SUMMATION PROBLEM
8-21]
215
and since this integral also converges uniformly corresponding to an
arbitrary s > 0, we can find Xq such that
I 1 < for iC > Xq.
Thus for X > Xq,
(2) li?’(:r|co)-Q„,(a;)j<£.
Thus we conclude that, for large values of x, F{x\ a>) is asymp-
totically represented by Q,nip)^
As an illustration, consider
L
Since X];
is uniformly convergent for ^ ^ 1, it is sufficient
to take m = 1, and we have approximately, when x is large,
(3)
X
§
1 A , 1 J
jA! = log»-g.
Again consider ^ log ^ A
0
Here taking m = 2, condition (C) of 8-15 is satisfied and we have
approximately, when x is large,
(4) log^A^= (^-4) log2;-a; + j^.
Evidently a grosser approximation is {x-\)\ogx-x, which
corresponds to the case m = 1.
Actually for m = 1 condition (C) is not satisfied, but
is still uniformly convergent.
Condition (C) is in fact a sufficient but not necessary condition for
the convergence of the integral, which is all that is actually required
in 8-15.
We can utilise the asymptotic property of the sum for values of
X which are not large in the following way.
THE SUMMATION PROBLEM
[8-21
216
From tlie definition it is clear that
1-1
F(x\(i>) = F{x+7m\ (£,)-(£> 2 9i(aJ+sco),
SO tliat
1
F{x\(F) = Q^(a;+?iw)-co 2
s=0
+ [F (a;-l- mco I co) - (a; + mco)]
n-1
= Qm{^)+(>^ S [A 6m (a; + SCO) -^(x + SCO)]
8=0 w
-{“ [-F {x + ^CO I ~ "1" ^0))].
If we now let n-^co , the term in the last bracket -> 0 by the
asymptotic property proved above, so that we have the development
00
(5) F(2:|<o) =6„(a;)-co S[^(a: + «co)- A6m(a: + sco)],
which is valid for c. We have also the equivalent form
n-l
(6) J(cc| o) = lim [Q^{x+n(x>)-o} 2 4^{^+soy)].
n-^co 5=0
8-22. Asymptotic Behaviour of the Sum for Small
Val ues of co. To study the behaviour of F(x\ co) for small values
of CO, we have from 8*21 (1),
CO) = 6m-i(a:)+|jF^<^<’"-i>(a;)+J?m(a;),
so that
CO— +i[F(a:| co)-6„_i(a;)] = to—
where
6i-’»+iP„_i(a:) = CO [^^(’»-i)(a;) + co-«P„(a;)J .
We now assume
i 1%
8-22] THE SUMMATION PROBLEM 217
to be bounded, less than A say. Since | 1 is bounded, less
than B say, we have
CO"
It follows that
is bounded, and hence
so that
lim R^_^{x) = 0,
<0 — >-0
lim [^(03 I w) - = 0,
w— >0
and hence, taking m = 1, that
z
lim X (i) A i = I ^ (0
ti) — ► 0 (i) j c
Moreover, if <^(cc) possess derivates of every order, and if an
integer tIq exist such that, for n'^n^,
is bounded, the above argument shews that
lim co-w+i [F (x 1 0)) - Qm-i (a;)] = 0
for every fixed value of m, such that m 1, and consequently
in accordance with Poincare’s definition of asymptotic expansion *
I CO) ~ d«+ S
Jc v=l Vi
for small values of co. Poincare’s definition is as follows,
series
A divergent
* Modern Analysis (4th. edition), p. 151.
THE SUMMATION PROBLEM
218
[8-22
in whick tke sum of the first (n+1) terms is is said to be an
asymptotic expansion of/(z) for a given range of values of arg2
if the expression R„{z) =z«[/(z)-fi„(2;)] satisfy the condition
even though
lim jB„(2) = 0 (n fixed),
\Z\-^CC
lim I 2?„(2) 1 = 00 (z fixed).
^->•00
When this is the case we can-make
where s is arbitrarily small, by taking | z [ sufficiently large. We
then write
/(z) 2 A^z-”.
n=0
In the case above, when o) — >0, oo .
We shall also use the symbol --- in a slightly different sense, which
will cause no confusion, as follows. Jjetf{x), g{x) be two functions,
such that
lira
X—>-ao
■IM
9{^)
is finite. We shall then write
fix) -~-g{x).
In our applications of this notatio.n the limit in question will
generally (but not invariably) be unity. In case the limit is unity,
we can say that f{x) and g{x) are asymptotically equal. Thus the
result of 8-21 (2) can be written
-f(a:lw) ~ Q,n{x),
and these expressions are asymptotically equal.
8*3. Fourier Series for the Sum. To obtain a Fourier
Series valid in the interval c cCq < a; < + co, we can proceed as
follows :
By 8*0 (6) we have the uniformly convergent series
poo w
■^(^1 w; p.) = ^ <f>ix+soi, p).
Jc fi=0
8*3]
THE SUMMATION PROBLEM
219
T V w 2nnx ^ . 2nT:x\
Put F(x\oy;[i) = + 2j [^71 cos --- + (3^ sm — - J ,
2 C^o+^ 2nnx j
a« = - -F(a;lco; t^)cos-— to,
O) Jxo ^
2 , . . 2mix J
Pn = - -^(ajlco; [x)sin-— (Za:,
JXo C)
and therefore
2 fa;o + ‘- 2^'^ ,
«» + *?«=- J?'(a:|co; [i.)e “ (ia:
Jxo
r^co+w
= -2
Jxo fi = 0
2nrrxi
e “ to.
Since the series is uniformly convergent we may integrate term
by term, so that
00 rxo-\-oi 2niTxi
«« + ^Pn = -2 2 (j>{x + so),y.)e “ dx
s = 0 Jxq
" «o+«»+u ?«™}
= -2^1 j>(x,\x)e “ dx.
5=0 Jxo+Sto
rx) 2mrxi
Thus oLn + i^n = - 2 <f>(x,]x)e to.
JX(i
Now when -> 0, J’ (x | co ; [ji) -> J’ (x | co).
If then, when fx 0, -> Pn
n/ I V 1 / 2^712; , . 2^710; \
i^(irico) = 1^0+ 2^ (a„cos--— +6„sm-— ),
n = l \ 0) CO /
r^o
ao = 2 j (ic) to,
= - 2 lim f ^(cc, jx) cos to,
= - 2 lim f ^(a;, jx) sin — ™ to.
)u,->-0 Jaro ^
. 2n'KX J
sin to.
CO
220
THE SUMMATION PBOBLEM
[8-3
Denoting the Fourier Series of F{x \ o>) by S{x), we have by a
imown property of such series *
2S{Xq) = /SI(a;o*-0) + /S(a:o+0)
= F{xQ + a j o4) + F(a;o | to)
= 2F (ajo I o) + CO ^ (a:o).
Thus
^(KoIco) = S(»o)-i-w^(a!o).
Writing x for x^ and noting that
f°° j /j \ / 2mzt 2mzx . 2mzt . 2Tcnx\ -
j ^(t,jj,)(cos cos hsm sin jdt
Jx \ 63 CO CO CO/
f°° I , , . V 2nnt ,
= I <56(0? + ^, pi) cos dt,
Jo CO
we have the following series which is valid for c,
F{x\o))= f <fi{t)dt-\ix^<f>{x)-2 2 f + pi) cos^^^^
J c /ut— >-0 Jo 0)
8*4. Complex Variable. Notation. We now proceed to a
discussion of the equation
(1) A«(2) = ^4(2),
iU
on the supposition that the variable and co may both be complex.
We shall denote the complex variable by z, x according to con-
venience, and in particular we shall write
= ^-h 47] =
CO =
The expression R (z) denotes the real part of z.
To avoid repetition we shall understand by e, pi arbitrary positive
numbers which can, in particular, be taken as small as we please.
The letters m, n, s will denote positive integers, while oc will denote
a positive number, such that 0 < a < 1.
When C = p where p is real and positive and where ^ is real,
we call p the modulus and ip the argument of 5^. We then write
mod ^ = I 1 = p, arg = ij;*
cha;^'^’* The^y of functions of a Meal Variahle (2nd edition, 1926),
THE SUMMATION PROBLEM
221
8*4]
The complex number = 5+^7] can be represented geometrically
by the point (I, 7]) referred to rectangular Cartesian axes or by the
point whose polar coordinates are (p, The figure which thus
represents ^ is called the Argand diagram and we can speak of the
point C It is easy to prove, and is in fact obvious from the diagram,
that
where numbers, real or complex and therefore that
Take a point a represented on the Argand diagram and surround
a with a small region, say a circle whose centre is a. This region
will be called a neighbourhood of a.
A function /(Q is said to be holomorphic in a region when f(Q
has a unique finite value and a unique finite derivate at every
point of the region. The function is said to be holomorphic at a
point, if a neighbourhood of the point exist, in which the function
is holomorphic. A point at which the function is not holomorphic
is called a singular point or singularity of the function.
Letf(Q be a given function, a a given point and N a neighbour-
hood of a. If in N an expansion exist of the form
/(C)
where g{Q is holomorphic in N, then a is said to be a pole of
order m of the function /(^).
The coefiScient r of (C,- a)“^ in the above expansion is called the
residue at a of the function /(Q.
If m = 1, a is a simple pole, and in this case
r = lim(C-a)/(C).
A fimction which is holomorphic in a region R except at poles, of
which every finite sub-region of R contains only a finite number, is
said to be meromorphic in R.
We now state
Cauchy *s Residue Theorem. Let Cbea simpleclosed contour,
such that a function is holomorphic at every point of C and in the
222 the summation problem [8-4
interior of C, except at a finite number of poles inside the contour.
where Si? denotes the sum of the residues of f(K) at those poles which
are situated within the contour C.
For the proof of this theorem and for a full discussion of the
subjects of the above summary the reader is referred to a treatise
on Analysis.*
8*41. Application of Cauchy^'s Residue Theorem. In
Fig. 1, .4 is the point ( - a, 0) JSC, ED are the lines yi = h,ri = -h]
CPD is a circular arc centre 0 which cuts the real axis at a point P
between n and n-^1 say, for definiteness, at the point n + ^.
Fig. 1.
As C describes the contour ABODE A, we shall suppose that the
point describes a contour which lies entirely in a region of
the plane in which the function <f>{z + (3^'Q is holomorphic. Since
Tt cot 71^ = p; + y — + y _ + . . . + y ^ y ,
* For example. Modern Analysis, 1, 5, 6.
THE SUMMATION PROBLEM
223
8-41]
it follows that the only singularities of the function
0(z+<o^) TtCOt TzC,
which lie inside the contour ABODE A are simple poles at the
points 0, 1, 2, ... , n.
We have therefore, by Cauchy’s Residue Theorem,
” If
X <i(z + SO)) = 1 ^(z + <ol^)7tcot7d^d^,
J=0 ^T^JABODjEA
since the contour is described clockwise.
Now
1 ^ „ 1, 1 _1 1
cot w; - 2 + 1 _ g-2Wf 2 1 - eSrir
Hence '
S. (-5+r:FST()^(»+“0‘«
ABDP
1
1^
2 1 -
<^(z + a)C)
= -i\ ^(z+6)C)d!:-j f </>{z+<A^)dK
J ABCP J AEDP
r «^(z+coQ,v f <^(g+<^Q jv
^ .Lbcp 1 - ^ ]abi>p 1 -
Since <j){z + co^) has no singularities inside ABODE A ^ we can
shrink the paths ABCP, AEDP in the first two integrals until they
coincide with AP. We now suppose further that, when X, describes
the contour ABFCPDGEA (obtained by producing AB, AE to
meet the circle CPD), 2:H-co^ describes a contour lying entirely in
a region in which + is holomorphic. We can then replace
in the second two integrals the path ABCP by AFCP, and AEDP
by AGDP.
Thus we obtain
” ^ 4>{Z'j- soy) = -- I <l)(z+ oyX,) dX, + 1
S — 0 J — CL O'
(^(Z+CaQ jy
+
^(g+O>0 , T
1 - ^27ri{ ^ n+h
224
where
THE SUMMATION PROBLEM
[8-41
Ln+i
f
JFCP
I _
<^(2:+6)Q jy
We now make the following hypotheses :
(i) For every n, however large, and for a fixed value of the
angle in Fig. 1, when ^ describes the contour AGDPFA,
describes a contour lying entirely in a region in which (5&(2;+coQ
is holomorphic.
poo
(ii) Tliat j <f>(K)d‘Z is convergent.
00
(iii) That 2 is convergent.
(iv) That I | 0 when n — > oo .
The principal solution of the difference equation of 8-4 is then
F(2j|co)= f 2 +
Jc s=0
Urns we have
(1) F{z\<^)
pZ - a>a p
^(Qc^!: + co
Jc JjRi
^(g + 0)C)
2
4>{z+ojK)
1 _ e2wi{
dL
Pia- 2.
8.41] THE SUMMATION PROBLEM 225
where Ri and are rays from - a to infinity, each inclined at an
angle p to the real axis, Fig. 2. If we now put
(2) m
we can write (1) in the form
F(2 I w) =/(z - «a)) + fiz+o:>Q dC
4"
f
Jjj 1 - e
d
^ii dz
Integrating by parts, we obtain for F (x | co) the expression
f 2Ttfe-2«f/(2 + coO f 2me^”if(z + coQ jy
Now when oo along R^, | \ -> oo , since R{ - iQ is
positive. Similarly, | | oo when >c3o along i?2- Thus the
values of the contents of the square brackets vanish at infinity.
Again,
' ’ =-i,
_ ^27ria
^Tli 2Tci 1
7U^
27zi sin^TT?^'
Thus we have
<’> ^('i“l = 25l/<^+“0C-Erk)‘‘‘^-
where C is the line of integration shewn in Fig. 3.
Rg. 3.
226 the summation PROBLEM [8-5
8*6. Extension of the Theory. Having established the
form which the principal solution of the difference equation
(1) Au{z) = <j>{z)
4A
takes under the hypotheses (i)-(iv) of the preceding section, we
now consider some cases in which these hypotheses are not fulfilled.
That such cases of exception are numerous and important may be
seen by considering such simple functions as sinh z,
When we are given an equation of form (1) where (j>{z) does not
satisfy the conditions of convergence enumerated above, we replace
the equation by
(2)
•a
and we then attempt to determine a function X(2^) such that this new
equation may have a principal solution of the form already found.
Denoting this principal solution by jP(2: | co ; p), we examine the
behaviour when fx~>0. If p) tend to a definite limit
function | co) which is independent of X(2;), we have the required
principal solution of (1).
To study this process, we consider the function (j>{z), which
satisfies the following conditions :
(A) In the half plane R{z)'^a, ^(2) is holomorphic.
(B) When R[z) > u, there exist positive constants C and k, such
that
however small the positive number s may be.
The class of functions which satisfy these conditions includes
aU integral functions of order * one and, in particular, all rational
functions.and functions of the form P{z)e^^ where P{z) is rational.
We shall now prove that in this case it is sufficient to take
X(2;) = p > 1, in order to ensure convergence of the integrals,
provided that co be suitably restricted.
We shall denote by S a real or complex number, such that | S | -> 0
when I ^ I CO . More precisely, given a positive number h, we
* For the definition of order of an. integral function, see, for exanaple, P.
Dienes, The Taylor Series, 1931, p. 290.
THE SUMMATION PROBLEM
227
8-5]
can find a positive number such that j S | < ji, if p = i 5^ | > Po-
If S occur more than once in the same formula, its value is not
necessarily the same in each place where it occurs. With this con-
vention we can, for example, write
X(2;-f6)Q = = (pa)2'e^p(’^+^)(l + S),
pP(l + §)^ > 1.
Too
Consider 1 taken along the real axis. We have
It follows from this that the integral converges since its modulus
is more convergent than | dp.
Again,
2 ^(2 + 6)5)
s =0
<C (7 g(Jfc+<) 0-8 - (Ai(er«)^ cos JJt) (14- S)
8=0
and
(A -h s) 0*5 - (pi {gsY cos px) (1 -h §)= - (p, {gsY cos px) (1 + S).
00
Hence the series is more convergent than ^ e~~\ provided that
8 = 0
cospT > 0, that is, provided that | | < ^ . Since p is arbitrarily
near to unity, this gives
(3) kl<J,
which is the first restriction on co.
Since the integral and the series both converge, it follows that
(4) jp (2 1 6) ; p) = f <f) (Q 6“^^ - CO 2
Jc 8 = 0
exists as an analytic fimction of z, provided that (3) be satisfied.
228 THE SUMMATION PROBLEM [8-5
Now consider Fig. 3. As describes the path C it is easy to see
that R{z+oi'Q^R{z- aco), provided that
5) P<|-|t|.
Hence, as ^ describes O', we shall have i2(2: + co?^) > a, provided
that a be chosen so that
(6) R(z-(x.co)'^ a.
We shall suppose conditions (3), (5), (6) to be fulfilled so that
condition (i) of 841 is satisfied. We now turn to (see Fig. 2)
With ^ on Bi, i; = p e»^ = p ei'*(l+S),
^ (sin^ 71^ + smh^
Hence
|ZijJ<Ce(*+Ol^l f (l + S)e-"(p)dp,
JjRi
where
w(p) = 27T7)-(A;4-e) po’+pL[(p(r)^cos^) (^ + t) ] (1+8).
The integral certainly converges if u(p) be positive, and this
condition is satisfied for all positive values of p, and for (i. = 0,
provided that
cosy(4^ + T)>0 and 2Tcpsin4^-(i+£)orp >0.
Since on we have 4^ > p and ip - p -> 0 when p ^ oo , these
conditions lead to
1 ^
271 sin A
that is, to
(7)
|t|.
_ 27r cos T
If these conditions be satisfied, converges, and, when p. *
■0,
^(Z+toQ jy.
8-5] THE SUMMATION PROBLEM 229
In the same way we may shew that, if conditions (7) be satisfied,
TT^ “ '
converges and, when (j, 0, has the limit
f nr
The proof that condition 8*41 (iv) is satisfied by the function
<^{z) presents no difl&cnlties and is left to the reader.
We have thus proved that the principal solution of
Aw(2:) = ^{z),
tti
when I <f>{z) | < for R{z) '^a can be put into the form
8-41 (3).
(8) +
provided that
Era. 4.
Fig. 4 illustrates what we have proved.
Expression (8) represents the principal solution for 2; in the half
plane R{z) ^ a, and co inside a circle whose centre is the point “ and
230 the summation problem [8-5
whose raditis is The particular contour C depends on the value
of 6) inside this circle.
When T = 0, oa is real and the contour C becomes a parallel to
the imaginary axis. We have thus, for co < j
(9)
From this we can at once draw the important conclusion that,
if <f>{z) be an integral function (of order one), that is to say, holo-
morphic in the whole plane (excluding the point oo ), (9) represents
2?!
the principal solution not only for co < ^, but for every co inside
27U ^
the circle | co | = -^. Also,
This last result is of great interest as it embodies the comple-
mentary argument theorem for the sum of an integral function of
order one (at most), the arguments z, (x^-z being called comple-
mentary.
With the notation of 8*16, we have, for example,
£,,(a;-<o I -co) = 5„(a;| co).
That is,
which, in the case 0 = 1, gives
the formula of 6-5 (8). It thus appears that the complementary
argument theorem of Bernoulli’s polynomials (of the first order) is
a particular case of the general complementary argument theorem
(10) , and is shared by Bernoulli’s polynomials in virtue of the fact
that they are principal solutions of the equation
A^i(a?) =
8*53] the summation PROBLEM 231
8* 63. The Sum of the Exponential Function. We have
by 8-5 (9), 841 (2),
Fig. 5.
If we deform the path as shewn in Fig. 5, we obtain
7- J_f
“ 2niJc 27ci J -a+i-^oo
where the integrand is the same as in (1) and C denotes the loop,
the straight parts of which are supposed to coincide with the real
axis.
The residue at = 0 of the integrand is co e'^^. If in the second
integral we write 1 for we obtain
7 = -coe^" +
a-f-iso
sin ttC
dK
-cae^' + e™" I + -
J-a-i=o Wtt;/
The last integral is equal to
-r:F5=si._^ = ^"'
1 CO I <
Thus
THE SUMMATION PROBLEM
232
[8-53
Regarded as a function of m the sum is meromorphic,* with simple
poles at the points, co = s an integer. The poles nearest the
origin are at ± so that the inequality stated for | co j is in fact
m
the best possible.
If we write - m for m, we get
6
8
CO e~
+
m
Combining these results, we get
CO sinh m{z- -|co) sinh me
coshm^AC= 2
. T mo)
sinh
m
8
sinh mJ^ A ^ =
CO coshm(g-|co) cosh me
sinh -
mco
m
We may observe that neither of these sums vanishes when z = c.
If we write im for m, we obtain
^ cosmCA^
^ sinm^A^
CO sin m (2; - |co) sin me
2
sin -
CO cosm(g--|Q)
2 . mco
sin-
cos me
m
8*6. Functions with only one Singular Point. Let<^(2:)
have only one singular point, at the point % = where is
finite. Then ^ (25) is holomorphic outside the circle whose centre is
the origin and whose radius is Let and a be two real numbers
such that % < cos O^, a>r^ cos
Then (f) (z) is holomorphic in each of the half planes,
jR{z)^a^, R{z)'^a.
* See 8-4, p. 221.
THE SUMMATION PKOBLEM
233
8-6]
We shall suppose that outside the circle radius
where C and k are fixed positive constants and e is positive and
arbitrarily small. Then, if 0 <C <o < 2tc / k, It (x) > ct — ««, we have
(1) «) =
where is the line through ~a parallel to the imaginary axis
described in the direction -ioo ~a to H-ioo - a. Provided that
R{x) > a, a can always be chosen to satisfy the above condition.
Now consider Fig. 6, where we have deformed the contour Li into
igj the lines AB, EF being straight, collinear, parallel to the
imaginary axis, and at distance I from it.
Si is a semicircle, radius 27r/^, centre 0, By taking the straight
portions DE, BC long enough and suflB.ciently near to the real axis
we can always arrange, for any fixed value of o) interior to S-^ and
for R (x) > a, that, as z describes L^, x + oyz describes a contour
to the right of which has no singularity.
In the extreme case, co = 2Tl:^ / k, the contour described by x+o^z
is JDg turned through a right angle with the origin moved to x. We
can now shew that
234 THE SUMMATION PROBLEM [8-6
In order to do this we must shew that the integral converges. To
do this it is sufficient to shew that
I f <f>(x + o^z)
is finite where P is an arbitrary point, l + on EF,
If y be the imaginary part of 2;,
I (1 - I = e-’^2/(sin^ 7tZ + sinh^ izyY^
< e~^y I sinh ny
I <j>{x+<jiz) I < C +(^*+«) 1 w jy
so that the modulus of the above integral is less than
J2/0
(A; -h «) I w 1 2/ ~ 2^2/
dy,
2rc
27U
and this is finite if | co | < or, since s is arbitrary, if | co | < ™.
Thus (2) is established.
Consider now
(3)
f{x) = ^^i^{z)dz.
R{c) > a.
Since % is a singular point of <^(2), f{x) is, in general, many-
valued. To avoid this we make a cut * in the z plane from z-^ to
*A cut is an impassable barrier. The variable may not move along any
curve which crosses this barrier.
8-6]
THE SUMMATION PKOBLEM
235
- 00 parallel to the negative imaginary axis (see Fig. 7). In the cut
plane f{x) is single-valued.
Changing the variable in (1), we have for R(x) > a, and for
0 < CO < 27r / yfc,
] fa; - wa +0)^00 —
(4) jF (cc I co) = -s — r— f{z) 7z^ cosec^ - {z-x)dz.
Now keep co fixed and move x to the left from R{x) > a
to R{x) <«!• When we cross the singular point we increase
F{x\<J) by
(5) P (a: 1 m) = ^(z-x)dz,
where is an infinite loop round z^ as shewn in Fig. 7.
Thus for R{x) <a^, 0 <o^ <2tz / k, we have
1 f-a+iX) / 7C
(6) =
By suitably deforming the path of integration in (2), we can
consider this result as established for all values of co interior to the
semicircle S-^ of Fig. 6, and for the modified path.
To evaluate P{x | co), we have from (5), on integrating by parts,
P{x\ CO) =-^^fiz)cot^{z-x)j^+^.^^ni>{z)oot^(z-x) dz.
Since cot — (s^-ioo - x) = -f-i, we have, if % be a simple pole
(7) P{x\ 0^) = -TziRi — v: R^cot (x-z^)
where is the residue at % of Evidently then
(8) P{x I co)+P(a; | - co) = - 2TciRi,
and from the definition (5)
P{x+o^ I co) = P{x I co),
so that P (a; | co) is a periodic function of x with period co.
To extend our results to values of co inside the circle | co | = 27r / A,
consider (4). Taking R{x) > a and keeping x fixed, let us make
236
THE SUMMATION PROBLEM
[8-6
ci) describe the path inside the semicircle S^, radius 2njk, in Kg. 8.
The path of integration then passes from via Zj to remaining
Fig. 8.
tangential to a circle centre x, radius | aw |. As soon as has passed
the singular point the integral on the right of (4) increases by an
amount equal to an integral round the infinite loop Z^, thus
(9) ?(^K) = P(,|«.)+ 1 f-^ /(«„..)
\Sm7ZZ/
for the path since, for this path, (5) still transforms in (7).
Fig. 9.
K, on the other Land, tee go from o> to coj by the path pg. Fig. 9,
meido tie eemieitol. 5,, radio. 2„/t, ^ pa,„. gj. ^
THE SUMMATION PKOBLEM
237
8-6]
the right of (4) again increases by an amount equal to an integral
round the infinite loop Zj, but now we must evaluate (5) for a
negative value of co, and then we obtain, for the path
(10) -F(a?|coi)
1 f-a+ioo / -rr \2
where P{x | coj) is still given by (7).
Thus jF(ci? I o) exists but is not one- valued when co varies inside the
circle | co | = 2tc j hm the co plane. If, however, we make a cut in the
6) plane (hut not in the z plane) from 0 to - 2ni j k, (a; | co) is one-
valued for CO inside the circle ( co | = 2tc / A: in the cut co plane and the
value of F{x \ co) for negative values of co, co^^ = ~ co, is given by (10).
Finally, if we now let x recede to the half plane R{x) < %, keeping
a fixed negative value of co, we arrive once more at (2), which, by a
simple deformation of the path of integration, is seen to be holo-
morphic for all values of co inside the circle | co | = 27r/i in the cut
CO plane and for all values of a? in the cut z plane. We have thus
obtained the analytic continuation of jP(a; | co) in the above regions.
li<j>{z), instead of having one smgularity, have a finite number n,
all at a finite distance, with residues fig? ••• ? can proceed
in exactly the same way and we shall arrive at similar results, the
function jP(a; | co) of (7) being replaced (for simple poles) by
n n
P(a; I co) = -Tii ^ S cot (a? - - .
s=i s = i CO
In the case we have considered above, (n = 1), the numbers
and a can each approach R(z^) as near as we please. If n > 1 this
is, of course, not the case.
We can now obtaia a generalisation of 8*5 (10) connecting the
su3ns for complementary arguments. We have from (10), writmg
-CO for co^, cr-co for x, and -1-z for z,
P(a?-co I - co) = 2mRx+R(x-oy | - co)
j
-l+tt+ico
-l+a-ioo
f(x+<jdz) — ) dz.
^ ^ ^ Vsm izz/
Using (8), and observmg that -l<-l + a<0, we have
(11) fi(a;-co I -co) = -P(a;| co)-hfi(a; | co).
238
THE SUMMATION PROBLEM
[8-7
8*7. An Expression for F(x|-o)). Consider
#.(.10.) = g ^(-.) A. =
- c
where
Tar+wz C-x — mz
/(a;+o)2:) = j (l>(~z)dz = <j>{z)dz.
Now we have
X
F{x\-<^) = § <t>{z) A . = 9{x-<^z) {-^)^dz,
e
where
rx-o>z
g{x-o)z) = J (j>{z)dz = -f{-x-\'(j^z).
It follows that
F(x\-(i^) = co).
If then in 8*5 (4) we take \(t) = we have, if co > 0,
-o>) = 1
=:lim jw 2 [ ^(2;) ,
/i— >0 I 5==0 J — 00 j
which expresses ^’(a; | - co) as the limit of a sum.
EXAMPLES VIII
1. Prove that
X
a
(z - 1) (2 - 2) ... (z - m + 1) A Z = ^ (a; - 1) . . . (a; - Ji) - i j5f (a).
2. Prove that
O A^
-1
z(z+l) ... (z+w) Ma:(a;+1)... (cc+w-l)
1 f“+i dz
nJa z(z+l) ... (z+n-1)'
EX. VIIl]
THE SUMMATION PEOBLEM
239
and by means of the identity
1
= 1 V J_
n\^o^ ’^sJx+s’
x{x+l) ... {x-\-n)
shew that the constant can be written in the form
n ^ o
3. Prove that
X
[ V ^ (2)] A ^ = § ^(2) Az-i\‘‘'"^<l>(z)dz.
<»> 2w to> o
4. Obtain the following expressions for the periodic Bernoullian
functions,
p (x) — ( — 1 U+i cos imzx
I O (27r)2v^4l ’
2(2v + l)! sin27rnx
'(27t)2’'+i ~riF^ *
5. Prove that
•f 0
y ^ V 2 (2 V 4" 1)! cos 2tc??-cc
-1-0 (27t)2»'+1 ^2k+1 ^
and deduce the expansion
- log(2 sin Tzx) = 2 ? 0 < a; < 1.
n«=l ^
6. Shew that
1
2V+1
(_1)V+1
(2it)2-+i
2(2v+l)!
r
*^0
•®2v+i(^) cotnzdz.
7. Shew that
1 “ *
240
THE SUMMATION PROBLEM
[ex. viii
8. Prove that
10. If <j> (x) be an integral function^ such that
where C is a fized positive constant and s is positive and arbitrarily
small, prove that
c
for all fimte values of co, real or complex.
CHAPTER IX
THE PSI FUNCTION AND THE GAMMA FUNCTION
In this chapter we consider the application of Norlund’s principal
solution to two special forms of the function to be summed, namely,
x-^ and log x.
The first of these gives rise to the Psi function, the second to the
logarithm of the Gamma function.
Both these functions play an important part in applications of
the finite calculus, and both possess great theoretical interest.
9*0. The Psi Function. This function is defined by the
relation
Taking x, co to be real and positive, we see that the conditions of
8*15 are satisfied and the function therefore exists. When co = 1,
we shall write
^{x) = ^(ccl 1).
We shall now illustrate the results of Chapter VIII by obtaining
properties of the function co).
9*01. Differentiation of the Psi Function. From 8*2,
we have
|-f(*l»)= §(-j)a*+i,
1
241
242 THE PSI EtTNCTION AND THE GAMMA FUNCTION [9-01
aad generally
Tliiis we obtain
A
8^
(-1)"-
(w-l)
ilr(n-l) (j;| K>1,
I 7h — i.
and consequently, using 8*1 (8),
Az (_i)n-i
1 0)) +
(n- l)(l + oc)”“
Witb tbe aid of this result and 8*16 we are now in a position tc
stun any rational function. For example, using 8*1 (8),
z
c
4.+£2*+g(l+(j^)A*+£(j + j^)*
= 6*
0
Any rational function can be expressed as tie sum of terms of
the t3q)es a a;”, 6(cc+p)“”, and can therefore be summed. This
summation property is one of the most important applications of
the function.
For numerical values of T^(a?+1), 'T^'(cc+1),
the reader is referred to the British Association Tables, Vol. I,
(1931), where they are tabulated under the respective names,
di-, tri-, tetra-, and pentagamma functions.
For integration of “T" (cc [ o) we have, from 8*1 (4),
1
6)
fa+w f
« 1
-dt — logic.
1 f
[See also 9-67.]
9-08] THE PSI FUNCTION AND THE GAMMA FUNCTION 243
9*03. Partial and Repeated Summation. As an example
of partial summation we have from 8*1 (7),
(C ^
1 1
Z X z+w
coJi CJ«„ UzCJ4„„
X
= I j* '^'^{z\(s:)dz- ^ ^'^^(2;4-co I o>) A^:-
Now
^ ’J'(z + C0 I w) = ^ + i ^ (z I 6)),
while, from 9*01,
--^'(x\oy) + l.
Thus we have
2§^^(z|o))A2
= ■^^^(a;(w) + <i)'^'(a;|co) — o> — {z | co) dz.
As an example of repeated summation, consider the equation
2 1
Aw(a;|co)
From 8*12 we have a solution.
§?A<= g(*-«-«)jAi
1 1
1 1
= (a;-o>) 'T^(a3) ct>)-(£c-|co- 1).
244 THE PSI EUNOTIOH AIND THE GAMMA FUNCTION [9.1
9*1. Asymptotic Behaviour for Large Values of®. Proin
8-15 (11), we have “
(1) ^(xjo)
Write
Then
where
-“I V Jo (£5+coi)’»+l®‘-
Q^(x) = log^- |;5v6)>'(-1)
V s=l
— j"
dt,
_ r* -Pm(-<) / X y
Jo Vaj + co^/
we now shew that Ii„(x) ->0 when ® -> 00 .
We have, integrating by parts,
r = rr=%^T - r
Jo x+at L(m+l)(x+c^t)Jo Jo (m+l)(®+coi)2
Since P„+i(- 1) is bounded, the integral on the right exists when
CO , and, moreover, both terms -> 0 when ® 00 . J fortiori
Thus we have proved that
®-{^(®|o))-0^(®)} = 0.
idt.
It Mows that asymptoticaUy,* in the sense of Poincare,
(2)
from which numerical values can be calculated.
In particular, for large values of x, we have
'T" (a: I w) ~ log x.
Hence
l“)-log(®+wco)} = 0.
* See 8-22, p. 217.
9*1] THE PSi FUNCTIOlSr AND THE GAMMA FUNCTION 245
Now, from the dej&nition,
(a; ( co) = ^ (ic + W6) I co) — 6)
n-l T
V' 1
ir+5co
= |log (a;+ nco) - o) (^+ nco | co) - log (a;4- ^^co) .
Thus, as in 8*21 (6),
^(fl;|a))= lim |log(a;H-^co)-6> ^ ^ — 1.
Putting ic — CO, we have
^(co(co) = logco+ lim |log(n + l)- 2
«~>oo I s ~0 ^ + t J
= logco-y,
where y denotes Euler’s constant.*
In particular, if co = 1, we have
^(1)= ~y.
We also note that the asjmiptotic series (2) is valid for all positive
CO however small, so that, when co -> 0 by positive values, we have
lim ^ (aj I co) = log x.
w->0
9*11. Partial Fraction Development. From 8*21 (5), with
m = 0, Qq{x) — log X, we have
% r 1 1
’S' (a: I o) = log a;- CO - A log (a:+sco) |
1 >)!
Putting cc = CO, we have, from 9*1,
logco-T = logco-2jji-log(l + j^)}.
Subtracting, we obtain
co)-logco + Y = - S
* Modem Aimlysis, 12*1.
[9*11
246 THE PSI FUNCTION AND THE GAMMA FUNCTION
where
^ *?o S + ®) “^og (s+2) + log (5+1) 1=0.
Thus we have
(a; 1 0)) = log 6) - Y~
00 >
V f- ^
s=o ^^~hO)S
This expression of Psi in partial fractions is valid not only for
a; real and positive, but in the whole complex plane, and shews that
^ (a:| co) is a meromorphic fimction of r with simple poles at the
pomts 0, -CO, -2co, -3co, ... , at each of which the residue is -co.
Regarded as a function of co (x fixed), we see that "P (r I co) has
poles at the points -r, -^x, -ix, - Jr, ... , and that these poles
have the pomt co = 0 as a limit point, so that co = 0 is an essential
Singularity.
9*2. The Multiplication Theorem. Prom 9-1 (1), we have
^(a;ico)-logco = log-- V lzi)L?r
w „~1 V \x/
Thus
+ (-1)™
dt=
(1) ’P(r|co) = ’p(£) + logco.
From 8*1 (1), we have
1
m
CO
m
= (mxjcd)~logm,
by a double use of (1). This is the multiplication theorem.
9-2] THE PSI EXJNCTIOlSr AND THE GAMMA FUNCTION 247
In particular, for m = 2, we have the Duplication Theorem,
namely,
^ (2CC I CO) = (il?| + {X+ I- I Cl>) }+log2,
and in particular, for 0 = 1,
^ {2x) = + ^ (a;4-|) } + log2.
Putting a; = I, this gives
^(i) = --Y-log4.
9*22. Fourier Series for ^ (a?). From 8*3, we have in the
interval Xq<x
{x) = -|ao + ^ [a^ cos 2Tcna; + sin ^nnx) ,
where
l{a^ + ib^) = - lim
Ju’o
-l:
00 g— a.r+2nTraji
dx
cos ^mzx
dx-i
p si
1 a’n
sin 2nTca:
dx
= ci {2mzXQ) + i si {^rnix^) ,
where ci(ir), si (a;) are the cosine and sine integrals, namely *
’ sin^
For Uq, we have
Thus
iW = ~<Z«, si(a;)=
1
-^dx = \ogx^.
J I X
'-dt.
■^'(05) = log oJq + 2 2 {ci (STcnajj) cos 2nnx + si [2nnx^ sin 2mzx) .
n = l
9*3. The Integral of Gauss for (a?). For J?(a:) > 0, ^ > 0,
we have, summing the geometrical progression,
o —
g~t(x+s) ~
* For numerical values of these integrals see British Association Mathematical
Tables, voL i, London, 1931.
248 THE P8I FUNCTION AND THE GAMMA FUNCTION [9.3
Integrating with regard to t from jji. (> 0) to 00 , we have
^ e-M<«+s)
x+s
Also,
/•qO gf—Xt
1
f” —dx = r ^ dt.
Ji a: t
Now, from the definition,
(»+«)■(
wldcli is Gauss’ Integral for {x).
Putting 2? = 1, we have
1
=r
J 0
9*32. Poisson’s Integral. As an application of 841, we have
fa; -aw 1
-dz
r~<x+ioo
{x + o)z) {l-e^^'^^) *
Put a = J, replace x by a;-h|-co and z hj z-^.
We then obtain
dz
: + ca
r -i«
Jo
dz
'$“(a;+|-co| 6>)
fix
-loga:+wJ^ (®+eiz)UTe^^^
In the first integral write z = it, in the second z = — it^ then
'^^(a;+|o)lco) = logic+ico f ( — ^
= logcc-f2<o2 I
Jo
which is Poisson’s Integral
' +6'
tdt
0 {x^+c^H^){l + e^^*)
9*4] THE PSI FUNCTION AND THE GAMMA FUNCTION 249
9' 4. The Complementary Argument Theorem. From
8*6 (7) and (11), we have
(a; “ CO I - co) = (a? I co) + 7ti+ 7t cot
Now by 8'7,
nx
CO ‘
r * .-M(a5~5a,)2 Cl 'V
(a; I “ co) = lim | co ^ dz i ,
^ ' ^-^ol .=0 a;~5co J^ao z J ’
while
( - a; I (o) = lim
By subtraction,
-.{i:
6“" , A
dz-oa X f .
C+oo
(a? I - co) ~ - a; I co) = - lim 1 dz = ni,
/u.— >0 j - 00 Z
the integral being taken along the real axis with an indentation at
the origin. Writing a; - co for x, we have
(a; - CO I - co) - (co ~ a; ( co) = ni.
Thus we have
(co “ a; I co) = (a; | co) H- tt cot ^ ,
which is the required relation between functions of the complemen-
tary arguments co, co - cc.
9*5. The Gamma Function, We start from Norlund’s
definition, namely,
(1) logr(a;) = ^ hgzi\z+c,
where the constant c is chosen so that log r(l) = 0. In order to fix
the determination of the logarithm, the complex plane is cut along
the negative real axis and the logarithm determined by log 1=0.
We have from (1),
A log r (a?) = log a;,
whence we obtain
(2) V{x-^l)=^xr{x),
which is one of the most important properties of r(a;).
In particular, if be a positive integer, we have
(3) r(n+l) = n!.
250 THE PSI FUNCTION AND THE GAMMA FUNCTION [9-5
Again by 8-1 (8), we can write
log r (35) = ^ log constant.
1
Differentiating tMs result by means of 8*2, we obtain
SO that (x) is the logarithmic derivate of F (x).
For numerical values of F (0?+ 1) see the British Association Tables,
VoL i, where the function is tabulated under the name x\.
9*52. Schlomilch^s Infinite Product. We have from 9*11,
r(a5) ‘ /ro^a: + s 1+s/
Integrating from 1 to a: + 1, since log F (1) = 0, we get
logr(35 + l) =
8 = 1 ^ ^
= log e-^* - S (log ^ - log e
8 = 1
Thus we have Schlomilch’s Product, namely,
(1) r(3: + l) = e-’* n —•
‘=^1+5
5
Since r(ir+ 1) = xV{x), we have
(2)
The infinite product in (2) converges absolutely at every finite
point of the plane, so that 1 / F (cc) is an integral function with simple
zeros at the points 0, - 1, -2, -3, ... . It follows at once that
F(a;) is a meromorphic function with simple poles at the points 0,
-1, ~2,....
The above product (2) was taken by Weierstrass as the definition
of the Gamma function.
9-53] THE PSI FUNCTION AND THE GAMMA FUNCTION 251
9' 63. Certain Infinite Products. Consider tte infinite
product,
P= TT (^+^i)(^ + ^2)
/-i(s + 2/i)(s + %)’
where = 2/i+2/2-
The product is then absolutely convergent * and, moreover,
e(2/i+ya-«i-»2)/» = 1.
Hence we can write
P =
Using Scfilomilcii’s Product we have, therefore,
” (g+ai)(g+a;2) _ r'(yi + l)r(y2 + l) _ r(yx+l) r(y2 + l)
.=i(s + 2/i) (5 + 2/2) a:ir(a^)a:2r(a:2) r(a;i + l) r(a:2 + l)’
provided that + ^2 = 2/i + 2/2*
In the same way we can evaluate
/=l(^ + 2/l)(^ + 2/2)-*(^ + 2/n)’
where + + ••• +^n = ^1+^2 + ••• +yn*
9*64. Complementary Argument Theorem. The infinite
product 9*52 (1) converges absolutely and uniformly in any bounded
region from which the poles are excluded.
Now
_x
V{l-x)^ey=^U " '
p — yx 00 pg
— nf^-
« 1 1+_
s
Thus we have f
r(-)r(i-) = in-^ = 3^,
1 ^
t Modem AnalysU, 7*5.
* Modern Analysis, 2*7.
252 THE PSI FUNCTION AND THE GAMMA FUNCTION [9-54
wMcli is the required relation between the functions of the comple-
mentary arguments x, 1-x, The result is originally due to Euler.
Putting a? = we have
9* 65. The Residues of T (x).
X = - n k Tn, where
= lim {x+n)T{x) — lim
x—^ — n x—^ — n
Now
The residue at the
{x-\-n)iz
sin TTic r (1 - x) '
pole
sinira: ' ' sin7i:(a;+w) -k tc
Thus
_ (-1)^ _(-!)»
using 9-5 (3).
We have therefore proved that in the neighbourhood of the pole
x:=-n the principal part * of F (a?) is P {F (x) }, where
■^^^(*)} ~ (i+w)wr
9*56. Determination of the Constant c. To determine
the constant c of 9*5, we have, from 8*21 (6), with m = 1,
logF(a;)-c= lim | [ log2;^J!2:~|log(a;+?^) - 2
n->*ao Uo «=0 J
Now, integrating by parts,
Cx+n
logzdz-\\og{x+n) = {x-{‘n~\)\og{x+n)-{x + n)
= {x+n-^) logn-n+(a?+n--|) log +
Also
(*+.-«log(l+D-» = («-l)log(l+^-J+^-...,
and this tends to zero when n-^oo . Thus
f n-l
(1) logF(a;)-c= lim ] (aJH-n-|) logn-r>7^“ ^ (a;+5) h .
W->ao I J
* Modem Analysis, 5*61.
9-56] THE PSI FUNCTION AND THE GAMMA FUNCTION 253
Thus, with cc = 1,
(2) logr(l)~c= lim |(7^^-|)logW“n-logn!| .
W->QO I j
Writing 2n for we have also
log r (1) “ c = lim I {^n + log w ~ 2w + (2n + J) log 2 - log {2n) ! | .
?i-^oo I J
Again, putting a; = we get
logr(i)-c= lim |n log w - ra - log - ^
J2,— >00 i. jU \
— lim |wlogw-M-log(2m)! + log(n!) + 2Mlog2).
Jl->00 L J
Adding the first and last of these three equations and subtracting
the middle one, we have
Thus
logr{|)-c = -ilog2.
c = log { V2 r(|)} = log s/2Tt:,
from 9*54. We have, therefore.
logr(a;) = ^ log2:A2J + log\/27c
as the complete definition of 9*5.
9*6. Stirling's Series. From 8*15 (11) we have Stirling's
series,
logr(ir + A) = log\/27rH-a;loga;--a?+5i(A) logic
^ v(v+l)a;*' mJo {x+z)^
This series is vahd not only for real x, but also for
- 7r+ s < arg a; < TT - s,
£ being arbitrarily small but positive.
Putting A = 0, we have
logr(£c) = log\/27r+(ic-|)IogiC“loge®+... .
254 THE PSI FUNCTION AND THE GAMMA FUNCTION [9-6
Thus
log r {x) ~ log (s/27c e~^) 0,
when I a; I cxD . Hence we have Stirling’s formula, namely,
Hm =
1 « 1-^00 ^27U 6“®
9-61. An Important Limit. Taking m = 2, in Stirling’s
series, we have
logr(a;4-A) = logJ^Tz-^- {x-{-h-\)\ogx-x+
^Jo {x+zf
Put = 0, and subtract the result from the above. We then get
Thus we have
I = 1’ 0<A<1.
\x\-^<a a;^r(cc) \ \
This result can be generalised as follows.
Let 5 be a positive integer, and let S denote a number real or
complex which tends to zero when ^ oo . The number S is not
necessarily the same in each formula in which 8 occurs.
By Stirling’s formula, we have
r(54-a;) = + e“^-^(l+S).
Hence
r(s + y) {s-^y)^+y^i^ ^ ^ ^
l+gj =«"(!+§), (1+fj =e-v(l + S),
l + tf*=l+S, (1+^
9-61] THE PSI FUNCTION AND THE GAMMA FUNCTION
Hence we have
Now I 5“+’*’ I = s“, when a, b are real.
Hence
r(s+a;)
r(s + y)
= s'(l+S),
where a = R(x)-R{y) = R{x-y).
255
9-66. The Generalised Gamma Function. If we define
the function r(a: | w) by the relation
(1) “logr(a:|(o) = ^ logzA^ + wlog.y'^^,
we have by diiferencing
(2) r(a?+co ( co) = xV{x\ co),
so that, if n be a positive integer,
r (n CO + 0) I co) = co" r (co I co) .
From 8*15 (11), with A = 0, we have
CO log r (a:; I co) = CO log + log x-x
— CO
= CO
m—l
B..
/coV CO
r Pmi^)
^x^ m
\ fx V
•dz
CO,
from Stirling’s series. Thus we have
(3) logr(a:|co) = log F i (a; - co) log co,
so that r (co I co) = 1, and therefore
r (nco “f CO I co) = co" n ! ,
when n is a positive integer. Also
r(a; I co) = r exp (i^logco).
256 THE PSI FUNCTION AND THE GAMMA FUNCTION [9-66
Again, from (1),
= log»-T-s/j:^-4T:)'
Integrating from a> to x+co, we have
o)logr(a;+co I co) = a;(loga>-y)~
whence
r(a;+ci) I co) = e H
a? 4-0)5 X
(j^s s
%
-‘(-+0'
^^50) /
and, by (2),
1 “ /
-=—> = e “ a; JI (l +
!|«) /iiV
-)
50)/
■ e
which shews that 1 /r(a;| co) is an integral transcendent function,
with simple Pieros at the points 0, - co, — 2o), - Sco, . . . , and there-
fore that r(ir | o) is a meromorj^ic function of x with simple poles
at the same points.
9*67. Some Definite Integrals. From 8-1 (4), we have
^cologr(2; I o))“0)log (^2: =J logzdz.
Thus we have Eaabe’s integral, namely,
1211
logr(2;| o))(f2: = a; log a; -a? 4-0) log y — »
and for x = 0,
I log r (2J I 0)) rfs; = o) log •
Again, from 9-66 (4), we have the integral of the Psi function,
namely,
I ^{z\<x>)dz = o)logr(x| co).
9-67] THE PSI FUNCTION AND THE GAMMA FUNCTION 257
From this and 9-3, we have, when w = 1,
Integrating, under the sign of integration, we obtain
f “ e-* / - 1 \
Iogr(a;+l) =J^ — (a3+y-— jdi, R{x)>-1,
a formula due to Plana.
9*68. The Multiplication Theorem. If m be a positive
integer, we have, from 8-1 (1),
2 { log r (x+ ^ - log s/27c| = log r
- log s/2m7z
= log r (mx) + ^ ^ log N/2m7u,
from 9-66 (3).
This yields Gauss’ Multiplication theorem,
iwi-i
V O TifhJC “ 5
T{mx) = {2Tc) ^ ■ U r(a; + ~).
s=o \ m/
In particular, for m = 2, we have Legendre’s duplication formula,
r(2:r) = 2^^-^'k-^T{x) r{xi-i).
9*7. Euler’s Integral for r(x). Subtracting (2) from (1) in
9*56, we have
f n-l 'I
logr(a?) = lim (cc- l)logn + log?^!”- ^log(^c+<5)f
JZ->00 I J
f n-l 'I
= lim ] xlogn+log(?2^-l)!-
n-i-oo I J
Hence
r(x) — lim
n^(n — 1) !
x(x+l) ... {x~\-n-l) *
Let t be a real positive variable and let log^ denote the real
logarithm of t. We dej&ne the many-valued function by
= exp (a; log ^).
[9-7
258 THE PSI FUNCTION AND THE GAMMA FUNCTION
Then, if R{x) > 0, we have
=
and consequently, from 2-11 (7), differencing with respect to a:, we
obtain
n ri
•^0
But A (^ - 1 )”.
Thus we have
)o a;(a;+l) ... (a; + rj)
Writing - for t, we obtain
1
do —
nl n®
Hence
a;(a;+l) ... (x-hnj *
r (x) = lim [ (l - -) "di.
. n-^co J 0 \ 91''
Thus we obtain *
r (a?) = f e”-* dt
Jo
which is Euler’s Integral for r(a;). This integral is known as the
Eulerian Integral of the Second Eiud
9-72. The Complementary Gamma Function. We give
the name Complementary Gamma Function to the function R (x)
defined by '
Fj (a:) = ~ e"* 0 ^ arg t^2n,
where i is the contour shewn in Fig. 10.
Fig. 10.
This contour consists of two straight parts ultimately coincident
with the positive real axis and an infinitesimal circle round the
* For the justification of this passage to the limit, see Modern Analysis, 12*2.
9-72] THE PSI FUNCTION AND THE GAMMA FUNCTION 259
origin. If R{x) > 0, the integral round the circl'e tends to zero as
the radius tends to zero.
Thus, if we start at infinity with arg ^ = 0, we have
ro
(a;) = -- er^ dt - e~^ dt,
Joo Jo
since is multiplied by after passing round the
origin. Thus we have
rj(aj) = V{x)-e^^^^T{x) = (1 r(£c),
and therefore
The loop L can be deformed in any manner provided that it
starts and terminates at oo and does not cross the real axis between
0 and 00. We now can write
^ 1
= ew_i
where the notation indicates * that the path of integration starts at
infinity ” on the real axis, encircles the origin in the positive sense
and returns to the starting point.
The above is HankeFs integral for r(jr). Although proved in the
first instance for R {x) > 0, the integral is valid in the whole plane
(since L does not pass through the origin) with the exception of the
points a; = 0, ±1, ±2, ... .
From 9-55, we see that near the pole a; = - of F [x), the function
ri(a;) behaves like
1 - (--1)^ _ _ 2iwiTzxe^^^ (-1)”
x-\-n n\ ~ x-^'U n\
Also
( - 1)" sin Tzx ___ sin 7c(a;+ ^^)
x~\-n ~ x+n ’
which is holomorphic at x = -n, so that F^ (x) is holomorphic at
the poles of F(a;). It follows that the complementary Gamma
function is an integral function of x.
* Modern Analysis, 12-22.
260 THE PSI FUNCTION AND THE GAMMA FUNCTION [9*72
Again,
ri(x+l) = = {I - xT {x) = xr^(x).
Consequently, ri(aj) satisfies tlie same difference equation as
r(a;), namely,
u{x+l) = xu{x).
9'8. The Hypergeometric Series.
the series
(1) 1+^®+
o(a + l)6(6 + l)
1.2.c(c+l)
a:2
This name is given to
a(a + l)(a + 2)6(6 + l)(6 + 2)
1.2.3.c(c+l)(c + 2)
where we assume that none of a, b,c is a negative integer.
Denote the coefficient of x'^ by Then when n-^oo,
(^ + ^) (^ + ^)
Un x^ (^4•l)(c+?^)
Thus the series is absolutely convergent if | a; | < 1, and divergent
when lx | > 1.
When j CP I = 1, we have *
n
where 0 denotes a function of n whose absolute value is less than
E I (where K is independent of n), provided that n be sufficiently
large. We conclude from Weierstrass’ criterion that the series is
absolutely convergent when | a? | = 1, provided that
jR( — cf *- 6+C+ 1) ]> 1,
that is to say, provided that the real part of c - a - 6 shall be positive.
Weierstrass’ criterion is as follows : |
* If be functions of the positive integer the relation
L=^0{z^)
means that an integer tiq and a positive number E independent of n exist, such
that 1 Cn 1<-^ ! I when n ^ Wq, See Modern Analyais^ 2*1.
t See K. Knopp, Theory of Infinite Series (1928), § 228.
9-8] THE PSI FUNCTION AND THE GAMMA FUNCTION 261
00
A series ^ of complex terms for which
where X > 1, and a is independent of n, is absolutely convergent if
and only if i?(a) > L
For B(ix)^0 the series is invariably divergent If 0
each of the series
00 00
Sl“«-W„+i| S (-!)”“»
n—0 n—0
is convergent.
If we denote the sum function of the above power series by
F{a,h; c^'-x), we infer that this, the hypergeometric function, is
an analytic function of x within the circle |a;l = l, and if
E(c-a-b) > 0, we have, by Abel’s limit theorem,*
(2) F(a, b; c; 1) = limF(a, b; c; x).
Gauss has proved the following relations satisfied by the hyper-
geometric function F{a,b\c\x):
{c-2a-{h-~a)x}F{a,b',c;x)-\-a{l-x)F[a-{-l,b\c',x)
-{c-a)F{a-l,h', c\ x) — 0,
c{c - 1 - {2c - a-b - l)x] F {a,h; c\x) + {c-'a) {c-b)xF{a,b\c+l\x)
~-c{c-l){\~x) F{a, 6; c-1; a?) = 0,
c(c4-l)-F(a, h\ c\ x)-{c^l) {c-{a + h-^l)x}F{a-\-l, 6+1; c-f 1; a;)
~{a+l){h-{-l)x{l-x) F{a+2, 6 + 2 ; c+2 ; a?) = 0,
each of which easily follows by considering the coefficient of in
the left-hand member. The verification is left to the reader.
9‘82. The Hypergeometric Function when x = 1. We
now prove that, if if(c~a~6) > 0,
F{a, b;c;
T{c)ric-a-^b)
^^T{c-a)T{c-by
* K. Ejaopp, loc. cit. § 100.
262 THE PSI FUNCTION AND THE GAMMA FUNCTION [9-82
We have, from the second of Gauss’
F(a c* D
relations, which holds when
F{a,b-, c+1; 1).
In this relation, write in turn c+ 1, c + 2, ... , c + w- 1 for c, and
naultiply the results. We then get
F(a, b; c; 1)
_ (c-a)...(c+9i-l-a)(c-6)...(c + w-l-h) ,
c (c + 1) . . . (c + M - 1 ) (c - a - -6) . (c + w - 1 - a - 6) ^ ^+'"5 !)•
Hence from 9-53, we have, when n oo,
provided that
F{a, b;e;l) =
r (c) r (c - g - 6)
r(c-a) r(c-h)’
lim F{a, b;c + n; 1) = 1.
>00
To prove that this is so we observe that | J'(a, 6 ; c + w ; 1) |
cannot decrease if we replace a,b,c by i a |, | 6 1, m - | c |, so that
\F(a,b-, c+w; 1)-1 1
^^{|g|...(|a|+s-l)}{|6|...(|6|+s-l)}
^*=1 a! (w-|c|)...(«+s-l-|c|)
< l«^l V" (l«l + l)---(kl + g-I)(|h| + lK..n6l4-,<?-n
-\c\^i a! (w+l-|c|)...(m + s-l-|c| •
Exactly as in 9‘8, we prove that this series converges if
«-|c|-|a|-|6| >0,
a condition which is always reah'sed if n be chosen large enough.
Also as n increases each term diminishes, and 1/(to - | c |) ->0 when
n^co. From this the required result follows.
9-84. The Beta Function. The Beta function is defined hy
B(a;, y) =
r(a:)r(y)
r(x+y) '
This function has the obvious properties :
B(a;, y) = B(y, x),
9-84] THE PSI FUNCTION AND THE GAMMA FUNCTION 263
B(a3 + 1, jr) y),
B{x, y+l) = ^-'&{x,y).
Differencing with respect to x, we have also
AxB(a;, y) = -^B(a;, y) = -B(a:, y + l).
If cc = 9^+ 1, a positive integer, we have
«/) = r(nH~ 1) r(y) 7^
’ iy+n)iy+n--l)..,yr{y) y(y+l)... {y + nY
so that, from 9*7,
B(^+l, ^)=: {l-tydt.
J 0
This is a particular case of the more general result,
Bix, y) = f R{x) > 0, R{y) > 0,
J 0
which we shall now prove.
The binomial theorem gives
5 = 0
^0 s\x{x+l) ...{x + s-l)
This series is uniformly convergent in the interval
0<e<i<z<l.
Multiply by and integrate from e to z. We then have
= V ^(a^+i)-(^+^-i)(i-y)(2-y)-(g-y) / ,+*_ 5+^^
^0 si x{x+l) ...{x + s) '
=^F{x, 1-y; 35+1; z)-^^’ (a:, 1-y; a;+l; s).
iC J/
264 THE PSI FUNCTION AND THE GAMMA FUNCTION [9-84
Since i?(a: + l - a; - (1 -y) ) = 2?(y) > 0, we have, from 9-8 (2),
]ixsiF{x, 1-y; a:+l; z) = F{x, l-y; jc+l ; 1).
Z-fl
Also since R{x) > 0, we have e® -->0 when s — >0, and the integral
converges. Thus, when s 1, s ^ 0, we have
= l-y; cc+l; 1)
1 0 ®
^ r(a:+l)r(a;+l-a;-l+y) ^ r(a:)r(y)
a;r(a:+l-a:)r(a;+l-l + y) ~ r(x-fyf’
where we have used 9-82. This is the required result. The integral
is known as the Eulerian Integral of the First Kind.
9'86. Definite Integral for the Hypergeometric Func-
tion. Suppose that | a: | <1. Then the binonoial series
(1 - xO-o = S 6(6 + 1) . .. (6+^-1)
s=0
is uniformly convergent for 0 ^ i ^ 1.
Multiply both, sides by where
jR(a)>0, R{c-a)>{),
and integrate from 0 to 1.
We then obtain, using 9-84,
[V-i(l-i)<=-»-i(l-a:0-*(*
^ 6(6+1). ..(6+s-l) r(a+s)r(c-o)
s! r(e+7)
g(g+l) ...(g+s-l) 6(6 + 1) ...(6+s-l) , r(a)r(c-a)
"o s!c(e+l)...(a+.-f-l)" ^
r(a) r(c-<2)
iW~~'
F(a, b; c; x)
= B(a, c-a) J(a, 6; c; x),
which expresses the hypergeometric function as a definite integral.
9*88] THE PSI FUNCTION AND THE GAMMA FUNCTION 265
9*88. Single Loop Integral for the Beta Function.
Consider the loop contour I shewn in Fig. 11.
Fig. 11.
We shall suppose that AB, CD coincide with the segment of the
real axis between 0 and 1 and that the radius of the circular part
tends to zero.
Now consider
/ = f C- 1)^-1
h Jo
with the notation of 9-72.
If we start with arg i = 0 along AB, we shall have arg - 1) = - tt
along AB and arg(i5- 1) = -f tc along CD,
Thus on AB, 1) = re“’-" = -r,
while on CD, - 1) = — - r,
so that if = 1 - r.
If R{y) > 0, the integral round the circle tends to zero when the
radius tends to zero, so that we have
= B(a;, y) = 2i sin B (cc, y).
Thus
1 r(i+)
B(^, y) = ^ dt,
^ 2^sln7uy Jo
Since F (y) F (1 - ^) = rr / (sin ny), we deduce from this the relation
266 THE PSI EUNCTIOH AND THE GAMMA FUNCTION [9-89
9*89. Double Loop Integral for the Beta Function.
Consider the contour shewn in Fig. 12, which starts from a point P,
passes positively round the points 1 and 0, and then negatively round
the points 1 and 0 and finally returns to P.
Fig. 12.
Consider
taken round the above contour. To evaluate the integral, we shall
suppose the contour reduced to four lines coincident with the
segment 0, 1 of the real axis, the radii of the circles round 0, 1 at
the Same time tending to zero.
If R{x) > 0, R{y) > 0, it is easy to shew that the integral round
these circles tends to zero when the radius tends to zero.
If we start at P on the real axis with arg ^ = 0, we have, for the
reduced contour,
On the path 1 : arg i = 0, arg(l - ^ = 0,
„ „ „ 2 : arg t = 0, arg(l - C = 2:1,
„ „ „ 3 : arg t = 27z, arg(l = 2tc,
„ „ 4: argif =: 27c, arg(l~C = 0.
Thus
Jo
= (1 - (1 ~ ^
Hence, with the notation of 9*72, we have proved that
(1 - e^’^) (1 - e>iv) B (X, y) = J
(l+),(0+), (1-). (0-)
p
9-89] THE PSI FUNCTION AND THE GAMMA FUNCTION 267
P being any point on tbe contour of Fig. 12, whicli may be de-
formed in any manner provided that the branch points 0, 1 are
not crossed.
The restriction E{x) >0, R{y)>0 can now be removed, since
the contour does not pass through the points 0, 1 and the above
double loop circuit integral gives the Beta function for all values
of X, y, neither of which is an integer. When either a; or y is an
integer the integral vanishes. We also note that
ri(a;)ri(y)
V{x+y)
Since Fj (a;), ri(?/), ^ are all integral functions, we see that
the above double loop circuit integral represents an integral function
of X (or y). We shall call the function y) the complementary
Beta function.
EXAMPLES IX
1. Find the sum from 0 to a; of the function
{x + 1 ) (x + 2) {2x + 3) *
2. Prove that
(i) (x) = log rr -
^0 (x + s
log ( 1 +
(ii) ^ (x) = log X -
1 j 2 (iC -f- 5) -f- 1
s-0 l2(ir-f-5)(cc+5+l)
-log
x + s+1}
x-i-s / ‘
3. Prove that
L If
prove that
(i)
9{x) = 2f:^-^,
(ii) g(x)+g{l-x) = 2% cosec ttx,
(iii) 5'(l) = 21og2, gii) = -x.
268 THE PSI FUNCTION AND THE GAMMA FUNCTION [ex. ix
5. Witli tlie notation of Ex. 4, prove that
1 2n_-l / \
6. Obtain the following results :
■j"w=;+i;
^0 (x + s)^(x+s+l)'
11*^ 1
S S) 2(x+s)^(x+s+l)^ ■
7. Prove that
8, Prove that
^(a:) = logir-A+2 j
[Legendre.]
tdt
0 (ir2 + 252)(l-e2.«)*
9. By means'of the last formula in 8*3, prove that
2 00
“vp (a?) = log a; - ^ H- 2 2 (2nnx) cos 2nnx
10. Prove that
+ si (27inx) sin 27rnx}.
(i) r(-i) = ^2VTu,
(ii) r(i)r(i)^2Tc/v3.
11. For large values of prove that, approximately,
3.5...(2n+l)_ 1 f , 3 1
2.4...2n t
12. Prove Wallis’ Theorem, namely
7u __ 2 . 2 . 4 . 4 . 6 . 6 . 8 . 1
2 1.3. 3. 5. 5. 7. 7. 9...’
and deduce that for n large
.r-i
n / ^{rcn) ’
EX. IX] THE PSI FUNCTION AND THE GAMMA FUNCTION 269
13. Prove that, if n be a positive integer,
1.3.6... (2w-l)
r(n + l)
2^
■ v^TC.
14. Shew that for real positive values of x the minimum value of
r{x) is 0-88560 ... , when x = 1-46163 ... .
15. Prove that
a; r(ir) r ( - x) = - 71 cosec nx.
16. Prove that
A (m + n-l) r(x)
+ (a:*- l)(;r-2) ... (a;- Ffa^ + m) *
17. Prove that
r(a7) =: 7(271)
X
fl + ±+.J L39
t 12a; 288a;2 51840a;3 2488320a;^ VxV/ '
18. By means of 8-3, shew that, when a;^ > 0 and a:^ < a; < a;j+ 1,
log r (x) = log s/2t: + Xg (log Xq-1)+ X) («« cos 2mix-\-bn sin 2n7:x)
where
+ ibn = 2 lim If -f{z) log 2; + [ ~ /(^) ^
/x->0 I L J:ro J
where
/(^■) = - f
J z
and hence prove that
exp {-\iz + 2niziz) dz^
Tcn = (log cCq) sin (2nnXQ) - si (27r7^^rQ) ,
7znbn = - (log Xf^ cos (2nTi:a7Q) + ci {2mxXQ) .
19. Shew that, in the interval Xq<x <XQ-i^l,
x-^ = Xq-~ 2 ;^sin27rn (x-Xq),
»=i
and hence prove, hy means of the last example, that
log r (x) = log 727r + (a? - 1) log Xq - Xq
- 2 — {si (2^^7l:a;o) cos (2^7ca;)-oi(2n7c2Cp) sin (2^1710;)}.
270 THE PSI FUNCTION AND THE GAMMA FUNCTION [ex. ix
20. From Ex. 18, deduce that
faro+l
27in log r (x) cos (2^^TO) dx = (log Xq) sin {^uxtiq) - si (27^3:^) ,
^ a*o
/* iCfl+l
27rn log r {x) sin (2^^7r5;) dx = - (log Xq) cos (2nTOo) + ci (27Tncr(j) .
•i (So
21. Prove that
i(a:)-logcc+ rL_^dz= lim [ fU
J 0 ^ n~> 00 LJ 0
f
^ 0
■ cos 2nz
dz
- log 9^Tr J
= lim
and deduce that
1 - cos 2nz 7 , , )
~ — . — - dz “ log in y ,
sin 2; & j ’
22. Use the last two examples to prove that
in I log r (cc) cos {2n7tx) dx = 1,
J 0
2Tcn f log r (cc) sin (2^710;) dx = y + log 2Tcn.
•'0
23. Prove that, for 0 < a; < 1,
logr(^) = log./2^?+ J {S2!^+(^ + Iog(2n,r))?^^^|,
71=1 L Zi/i j
and deduce Kummer’s series, namely,
log r (x) = (y H- log 2) (^ - a;) + (1 - x) log 71 ~ log sin tux
+ 2 sin 2^^7Ia;.
24. Prove that
B (a;, 1 - a;) = Tc cosec nx.
25. Prove that
B(a;, y) B(x-t-^,z) = B(^y z) B{y + z, x).
26. Prove that
r (a?[a>) r {a)~x\a)) = (w/a>) COSec (wx/cu).
[Euler.]
CHAPTER X
FACTORIAL SERIES
In tliis chapter we develop some of the properties of the series
X a:(a;+l) a:(a:+l)(x+2) a:(x+l)(x+2)(x+3) ’
which is known as a factorial series of the first kind, or series of
inverse factorials ; and the series
I I A (®-l) I 63(x-1)(x-2)(x-3)
which is variously called a factorial series of the second kind, a
series of binomial coefficients, or Newton’s (interpolation) series.
The last name arises from the fact that Newton’s interpolation
formula 3-1, when the series is extended to infinity, takes the above
form.
Both these factorial series have many properties in common, in
particular when x is a complex variable, the domain of convergence
is a half-plane.
Factorial series are of importance in the theory of linear difference
equations, where they play a part analogous to that of power series
in the theory of differential equations.
With the notations of 2-11, we see that the series can be written
in the respective forms
ao(x-l)(-i) + ai(x-l)<-2)l! + + ...,
6o + 6i(x-l)W/l! + 62{x-l)(2)/2!-i-...,
in which shape they present a marked analogy with power series
in x-1.
271
272
FACTOBIAL SERIES
[10-0
While factorial series first appear in the work of Newton and
Stirling, their systematic development on modern lines is due
largely to Bendixson, Nielsen, Landau, Norlund and Bohr.
The present chapter is based mainly upon N5rlund’s Legons sm
les skies d' interpolation (Paris, 1926), to which the reader is referred
for a more detailed treatment.
10*0. Associated Factorial Series. With the sequence of
coefficients
a-Q, <^25 >
we can associate the factorial series
(1)
(2)
2
a.. 5!
which we shall call associated factorial series,
fundamental theorem due to Landau.*
We now prove a
Theorem I. Associated factorial series simultaneously converge
or diverge for every value of x which is not an integer.
In the first place, suppose that (1) converges.
Put
We have
Thusf
x{x+l) ... (iC + 5)’
5! 5!
C, = X [I
smTTir
Um Cs =
5— >00 rc
Hence a constant K exists such that | I < K.
* E. Landau, Sitzsher. Ahad, Miinchen, 36 (1906), pp. 151-218. Landau also
proves that the Dirichlet series
formed with the same coefficients
theorem.
can be included in the enunciation of the
t Modern Analysis, 7 *5.
10-0]
FACTORIAL SERIES
273
and consequently
I *^s+l I i
-mi 1)2.
and hence the series S (Cg - is absolutely convergent. By hypo-
thesis, S bg converges. Hence by du Bois-Reymond’s test E bg Cg
converges, that is to say (2) converges.
Secondly, suppose that (2) converges.
Put
dg — { — 1)® ag
As before,
> fs — ^
{-lysisi
x{x^- 1) ... (x^-s^) '
lim/,=:^-- ,
5_>oo sinnx
so that a constant K exists such that |/s [ < K. Also
x^
fs - fs+l = fs •
Hence '^{fg-fs+i) converges absolutely and therefore we again
infer that E dgfg converges, that is to say (1) converges.
10*02. The Convergence of Factorial Series. Let
_a;o(a;o+l)...(a;o + g)_r(a;o + g+l)r(a;)
1) . (2/ + 5) r(2J + 5-1- 1) f (^Cq) ’
/o\ ,, _ (a;-l)...(a:-5) _ r(s-a;-l-l)r(l-^)
- (X, - 1) . . . (5o -“«) ~ r (5 - 2=0 + 1) r (1 - 2;) ’
and let a denote the real part of a; - which we take to be positive.
Then, from 9-61,
|MJ=S- (1+8).
r(l-^o) n+S)
where S — > 0 when s ->oo .
See p. 274 for the statement of this test.
274
FACTORIAL SERIES
[10-02
In the case of we suppose that x is not one of the numbers
0, - 1, -2, ~ 3, ... 5 and in the case of that Xq is not one of the
numbers 1, 2, 3, ... .
From these results it is clear, when x and are given, that both
and Vs are bounded.
Let Ws denote either Us ot'Vs*
Then
It follows from Weierstrass’ criterion (9-8), that
(i) when 0 < o ^ 1,
S («'s-%+l)
s = 0
is absolutely convergent.
(ii) when a > 1,
fi = 0
is absolutely convergent, and, a fortiori,
2 K-tOj+i)
is absolutely convergent.
CO
Now let 2 be a (not necessarily absolutely) convergent
series.
We have du Bois-Eeymond’s test,* namely that
S as hs is convergent if S (6^ — hsj^fj he absolutely convergent and if
S Us converge at least conditionally.
If follows that
00
2
«=0
IS convergent.
Moreover, Ws is bounded, and in fact we have
I Ws \ <Cs~^,
where C is independent of s. Also | | is bounded, since S ols
converges.
* K. Kjiopp, Infinite Series, 184. The test is substantially due to Abel and
IS also known as Abel’s Test.
FACTOEIAL SERIES
275
10'02]
Tims we have
m 1
2 las M’s
where C is independent of 5.
Therefore if (t>1, converges, and consequently
converges absolutely.
If we take :
(A)
a, . 5 !
we have
(B)
as = ( - 1)® a, ^), w, = u.
we have
We therefore have the following theorems :
Theorem 1 1. If a factorial series converge for x = the series
converges for every x, such that R{x) > R{Xq).
Theorem HI. If a factorial series converge for x = Xq^ the series
converges absolutely for every x, such that E(x) > R(Xq+1).
Theorem IV. If a factorial series converge absolutely for x = Xq,
the series converges absolutely for every x, such that R{x) > R(^q)-
For in this case S | | converges and | | is bounded so that
m m
^ I I { >
s~n s—n
where M is positive and greater than every | |.
10*04. The Region of Convergence. We can now prove
that the region of convergence of a factorial series is a half-plane.
For we can divide all rational numbers, excluding 2ero and
positive and negative integers, into two classes L and R, such that L
contains all numbers which make the series divergent and R contains
all numbers which make the series convergent. From Theorem II
we see that each member of L is less than every member oiR.
276
FACTORIAL SERIES
[10-04
The above classification therefore determines a Dedekind section
of the rational numbers and therefore defines a real number X,
such that the series converges for cc = X + e, where s is positive and
arbitrarily small, and diverges for x= X-s.
This number X is called the abscissa. of convergence. By Theorem II
the series converges for every point in the half-plane which is
limited on the left by R{x) = X. Theorem I can now be stated in
the form : two associated factorial series have the same convergence
abscissa.
Of the classes L and R by which X is defined one may be empty,
that is, may contain no members. If L be empty, we have X = + oo,
that is, the series is everywhere divergent. If R be empty, X = - oo ,
and the series converges in the whole complex plane. In both cases
the integral points are possibly excluded.
10*06. The Region of Absolute Convergence. The
region of absolute convergence is likewise a half-plane. If in the
definitions of the classes L and R of 10-04 we substitute the words
absolutely convergent ” for convergent,’' the Dedekind section,
by Theorem IV, determines a real number p. called the abscissa of
absolute convergence. The series converges absolutely in the half-
plane limited on the left by R{x) = p.
From Theorem III we conclude that
0<[x-X<L
In the strip defined by
X<R{x) < fjL,
the series converges, but not absolutely.
We now proceed to determine the value of X, but before doing so
we investigate some preliminary results.
10*07. AbePs identities. The two following identities, which
are due to Abel, are of frequent use in the transformation of series.
(I) If As = «^« + cj„+i+... + a5, then
m m
2 ^ ”■ hfi+l) “ ^3) m-
s=p
10-07]
FACTORIAL SERIES
277
(II) If — <3^5 + <^s+i “f" ^s+2 ■f' • • • 3 theii
m m~ 1
2 iKx - h) + K - 6m ^'m^X-
s—p s=p
To prove these identities, we observe that
~ (^S ■” ^S+l) (-^S ~ -^S-l) “t ^s-hl(^s ~
(^s ^s+l) ^s-1 + ^ S J
and that
~ (^S+1 ~ ^s) ^ S+l s'” ^S+1 ^ S+1*
From these results the identities follow at once by summation.
10*08, The Upper Limit of a Sequence. Consider a
sequence of real numbers,
(^n) ~ ^l5 ^3’ * * • *
Divide all rational numbers into two classes L and R, such that
if Z be a member of the class L there is an unlimited number of terms
Xn, such that x^ > Z, while if r be a member of the class J?, there is
only a finite number of terms x^ such that x^ > r. It is clear that
each Z is less than every r, and this classification involving, as it
does, all rational numbers, therefore determines a Dedekind
section of the rational numbers. This section defines a real number
X, such that, if e be an arbitrary positive number,
(i) > X - £ for an infinite number of
(ii) > X + £ for a finite number only of x^.
The number X defined in this way is called the greatest of the
limits, the upper limit, or limes superior of the sequence [x^), and we
write *
X = lim sup Xn^
n-->oo
In this chapter we shall be concerned with sequences whose 9^th
term is of the form
Iqg^^
log^ ’
* See K. Knopp, Infinite Series^ p. 90. Also Modern Analysis, 2*21.
278 FACTORIAL SERIES
where is real and positive. Let
[10-08
X = lim sup
log^n
'<X+s, if
' > X ~ £,
l0g9^ ’
and let s be an arbitrarily small positive number.
Then we can find an integer tIq, such that
^Og^n
log 5
log^n.
logn
for an infinite number of increasing suffixes where
% ^ 71q,
Thus
Xn < n^'^% if
Xn>n^~% if n = % ^2,
If a real number o, other than zero, exist, such that
lim ^2 = 0,
we find a suffix. iV, such that
iz?,; < £ if n'^N,
Hence if -n.^ be the first of 7%^,
such that > N,
that is to say,
we have n^-‘<en<^, if n = n„ ,^,^2.
- <r - f ^ 2 for these values of n.
Hence we must have X - cr ~ e < 0. Since e is arbitrarily small,
we have
X<cy.
Again, if Xn be such that
X,
n-^oologn
we have in the same way »>'-«< a;„ < e log w, if w = n„ ....
Thus log m' < which necessitates X<2s, and since
e is arbitrarily small, we must have
lim = 0,
X<0.
FACTORIAL SERIES
279
10-09]
10*09. The Abscissa of Convergence; Landau’s
Theorem. The convergence abscissa of the associated factorial
series 10*0, (1), (2) is determined by the following theorem.*
Theorem V. Let
OL — limsnp log ^ logn, p = limsup log 2 ^ logn.
s=0 8-^n
The abscissa of convergence, X, of the associated factorial series is
equal to oc 4f X'^0 and is equal to ^ if X<0.
We consider the series
(1)
.9 = 0 \
x-1
and divide the proof into four stages.
Suppose that the series converges at a point x, where x is not a
positive integer. Let R{x) = a.
(i) We prove that if o ^ 0, then a o ; and consequently that
a<X, if X>0.
Write
b, = (~iya.
x-l
(-l)^g! ^T(s + l)T{l-x)
(x-l) (x-s)~' r(5-a?+l)
Then we have
SO that, from 9*61,
c, = r (1 - a;) s- (1 + S), - c, = ^ r (1 - ^) 5-1 (1 + S),
where | S | -> 0 when 5 — > oo. Thus we can find a positive number K
independent of s, such that
1^3 1 <Ks<^, <Z5-1.
Let jBs = 63 + 65^^ + 634.2 -f . . . . Since as = 63 c^, we have by AbeFs
Identity
m-l
^2 (^3+1 “■ ^s) ^s+l "t ^3? -®!P “®W+1 >
s=:p 8~p
*E. Landau, Sitzsber. Ahad. MUnchen, 36 (1906), pp. 151-218.
280
and hence
FACTORIAL SERIES
[10-09
ub m ~ X
^ ^s+l~^s ^ ^s+1 d- Cj, Bp + C^B^
!s=j? i S—p
By hypothesis, E converges, and hence, given e > 0, we can
find 'p, such that | 5^ | < e, if a > Hence we have
(2) A^^pKeKi 2 +
^ s—p /
P+1
NowJ x^-^dx lies between s'^-^ and (s+l)‘'-i, and hence
Cm - 1 m- 1
x^-^dx lies between 2 + 2
•'P S-p s~p ^
whence we easily conclude, if o > 0, that
m-l
s—p
and hence
7n^ \m^
Now let m -> 00 . We then have
lim = 0,
m-^oo
and hence, from 10-08, a c.
Again, if cr = 0 and s<x<s+l, then
p+ifc.
Jjf X '
so that
^ ~ 1 m “ 1
2 5“^>logm-log^> 2
t^p m f
and hence, if p > 0,
m-l
2 < log
FACTORIAL SERIES
281
10-09]
Thus, from (2), we have
Am,],< sK{2+logm),
whence
^ < ^-szk 0 + g K f + 1 ) .
logm logm Vlogm /
It follows that
Hm ^21^0 3,0,
logm
and therefore again, by 10-08, we have a < a. Thus (i) is established.
(ii) We now prove that when a is finite the series converges for
X = QC+E, where e is an arbitrarily small positive number, and
consequently that X^oc.
Let
Then, by 9-61,
r{s-x-i- 1)
rT^TiynT^^*
r(r — rr) 2;)
so that if a; = a + £ we can find a positive number K, independent
of 5, such that
I ^5 1 <
I ds-ds+i I <
and further by 10-08, such that
Let
2 Cln
n~0
As — (iQ + a-^-\-ct^ + (Xs*
Then, since we have by AbeFs Identity,
«=37
Thus we have
m
{ds “ ^s+i) dp Ap^^ A ffi .
(3)
f V}
s—p t s—p
pcL + ie
^a + ie
282
FACTORIAL SERIES
[10-09
The right-hand member 0 when p oo , and consequently 2
converges. Thus (ii) is established.
Combining (i) and (ii) we have, if X > 0, a < X, and when a is
finite, X < a. Consequently we must have X = a, if X > 0.
(iii) We now consider the case a <0, and prove that if g < 0,
then ^ ^ a; and consequently that P ^ X.
When G < 0, we can let m~>co in (2), which then gives
-4oo.®= < eZ ^ .
S—p s—p
Now, if + and g < 0, we have
p+i
j s
Thus
s=p ^
Hence
p-‘'Aa,,p < eZ(l--+p,
SO that
hm —
p->ooP^
s=p
= 0,
and therefore, from 10-08, we have p < o, so that (iii) is proved.
(iv) Lastly, we prove that when p is finite the series converges,
for a; = p + £, where e is an arbitrarily small positive number, and
consequently that X ^ p.
Let
Ag — O5 + ®s+l + ®s+2 + • • • •
Then by Abel’s Identity we have, with the notation of (ii),
m 1
^ ~ ~ + ^'m+l ,
and we can now find Z, such that
00
n=s
10-09] FACTORIAL SERIES 28^
We thus get an inequality of the same type as (3) and we conclude
in precisely the same way that the series converges.
Combining (iii) and (iv) we have, if X < 0, then p < X, and when
P is finite, X < p. Thus if X < 0, we have X = p.
Another way of stating the theorem is the following :
\=z oL if the series 2 diverge^ and X = p ^ the series S con-
verge.
10*091 . Majorant Inverse Factorial Series. With the
notation of Theorem V, we have
a= X, if X>0,
a = 0, if X < 0,
for in the latter case 2 converges.
Hence if X' denote the greater of the numbers 0, X, we have
X' = a, that is to say,
lim sup log 2 ^logn = X'.
Hence from 10*08, given s > 0, we can find Uq, such that
n
5 = 0
< n ^ Uq.
I 5 — 0 I
Now, from 9*61,
/'X' + s + n\^ (X^ 4- £+ 1) (X^ 4- £ + 2) ... (7J 4“ £ 4”
n nl
— r(X"4-£4-y^4-l)
r (fi 4“ 1) r (X^ 4" £ 4“ 1) r (X^ 4- e + 1)
Hence we can find a positive constant M, independent of n, such
that
s«,
, 5=0 . \ n /
for all values of n.
Now consider the series *
(1)
_M M{X' + e) Af(V-h£)(V4-£ + l)
x-X'-s X x{x-hl) x{x-\-l)(x+2)
which is absolutely convergent, for R{x) > X'4-£.
* This series is obtained in 10*2, example 2.
284
FACTORIAL SERIES
[10-091
The stk term of this series is
x{x^l).,.(x-^sy 5 /•
If we call the the coefl&cients of the series, all the coefficients
are positive and the sum of the first ^ + 1 of them is given by
since this sum is equal to the coefficient of in
Thus, whatever the value of n, the sum of the first n+1 coefficients
of (1) is greater than the modulus of the sum of the first n^l
coefficients of the series
/o\
We shall call the factorial series (1) a majorant series for the
factorial series (2).
-f- £ +
10*1. Series of Inverse Factorials. We shall now con-
sider the function defined by
^oX(x+l)...(x+s)'
The region of convergence has already been shewn to be a half-
plane limited on the left by the line
Ji(x) = X,
where X is the convergence abscissa.
Since terms of the series become infinite when a; = 0, -1, -2, ,
we shall always suppose that such of these points as may lie in the
half-plane of convergence are excluded from the region by small
circles drawn round them. Unless X = - oo it is evident that only a
fimte number of these points can he in the half-plane of convergence.
10*11. Uniform Convergence of Inverse Factorial Series.
We shall now prove the following theorem due to Norland : *
* Series dHnterpolation, p. 171.
FACTORIAL SERIES
285
10-11]
Theorem VI, If a series of inverse factoriah converge ' at the
point Xq , the series is uniformly convergent when x lies in the angle A,
vertex at x^ , such that
- I-Tt + 7] < arg (a; - sJo) < - tq ,
where y] is positive and arbitrarily small.
Let
_Xq{x^+1)...{x^ + s)
x{x+l).,.{x + s) ’
R{Xq) = ao, arg(x-a;o) = 0, = r.
It is clear from Kg. 13 that we can find a positive integer n, such
that
a = I arg(a;Q + 5)| <-|y], if s^n.
Denote by P, Pq the points + Then the length of OP
is not less than the length of the projection of OP on OPq, so that
\x^~s\'^ |cco + 5|+rsin(|-7i:--04-oc).
Hence, if s^n,
(1) . \^+s \ > |a;o+5|+rsin^7],
286
■whence we obtain
FACTORIAL SERIES
[10-11
(2)
+ g ^ |a;o + g| cosa ^ gp + g
x + s +s|-fysin|7) cosa ^ cr^-f s + -|-y sinv] ^ ’
(3)
a:-a;o
a: + 5+l
<
cCfl + s+l |+rsini->] ^ CTo + s+l + |rsin7) ’
since | sin r] < sin ^-q.
Now,
and by (2),
Ms
= u
n~l
(Xg + n)
(x+n) ,
(a?o + ^)
(Xq + s) ’
(Xg+n)...(xo + s) ^ (gp + n) ... (on + s)
(a;-fw) ... (a;+s) (gp + M + irsinT)) ... (gp + s + lrsin-/)) ~
say.
Also when x lies in the angle A, u„_i is clearly bounded since n is
fixed and the points, -1, -2, ..., are excluded by small circles
dra'wn round them.
Suppose that | [ < N. We have then
lu,l<KUs, U,<1,
so that the are uniformly bounded in A.
Again,
Hence, using (3),
“s-Ms+l
= M,
a7-a;e
a:+s+l ■
(4)
If
r ^ ^
gp+s-fl+l^rsin'?] sin-ir;
{U.-U,.
^ a !
a'o(a^o't 1) (ajp-Ps) ’
the series S converges by hypothesis.
Hence, given s > 0, we can find N such that, if p > W,
S^s l<
ii= 'n
ssinTj
"W •
FACTORIAL SERIES
287
10-11]
Now, if jBg = + + we have from Abel’s identity,
if
m-l
S
s~p
< S l-SJ l«5-“ml + l-S
S = p
w I
8 = p
<sUp<e,
so that the series S converges uniformly in A, that is to say,
n{x) = I,-, r,
^ ' x{x+l) ... (ic4-5)
converges uniformly in A.
It follows from this theorem that the sum function of
the series is an analytic function of x in any closed region, which,
together with its boundaries, is interior to the half-plane of
convergence, for any such region can be enclosed in an angle of
the type >4. That the region of uniform convergence is even more
extensive than that indicated by Theorem VI is shewn by
Theorem VII. If the series of inverse factorials he convergent
for X = the series is uniformly convergent in the_ half -plane
R{x) = J?(a;(j) + £,
where z is positive and arbitrarily small.
It is clearly sufficient to suppose Xq real. Taking x^ = <Jq, and
n an integer such that n-f<7o>0, we replace inequality (1) by
\x+s\'^\ 0*0 + 5 1 + £, and sin rj by s in the remaining inequali-
ties, and the proof is then entirely on the same lines as that of
Theorem VI.
Thus Q(x) is an analytic function at every point within the
half-plane of convergence, with the exception of those of the points,
... , - 3, - 2, - 1, 0, which may lie within this half-plane.
10*13. The Poles of 0 (x). The function fi(a7) can be written
in the form
Q(x) = r(x)
a^s!
/torix-hs+iy
288
FACTOBIAL SERIES
[10-13
Since IjTix) is an integral function, we see that has simple
poles at such of the poles of r(a;) as lie within the half-plane of
convergence.
If a; = ~ n be such a pole, we have from 9-55 the residue of Q (rr)
at this point, namely,
(-1)" V
n\ g4o r(5-*n-fl)
00
(-1)” s
s~n
since 1 /^(5-7^-l-l) is zero, for 5 = 0, 1, 2, ... , n- 1.
If X = - 00 the function 0 (x) is meromorphic in the whole com-
plex plane.
We may note that, in terms of the Beta function.
00
8 — 0
10*15. The Theorem of Unique Development. A func-
tion which can he developed in a series of inverse factorials can be so
developed in only one manner.
For suppose that the same function admits two distinct develop-
ments. Then we have an equality of the form
^ - ^
Let X, X' be the abscissae of convergence. Multiply both sides
by X and let a; oo in such a way that R [x) — > oo in the half-plane
R{x) > X, jK(cc) > X'. We then obtain
^0 ~ ^0*
Eemove corresponding terms and then multiply by x{x+l) and
let R{x) CO as before. We thus get Proceeding in this
way we see that the two series are identical.
It follows from this that an inverse factorial series cannot vanish
identically unless all the coefficients vanish.
10-2. Application of Laplace's
10-13,
Q{X)= f] Mi =
s=o 3;{x+l} ... (k+s)
Integral. We have, from
00
2 ffisB(a;,5+l).
8 = 0
10-2] FACTORIAL SERIES 289
Thus using the integral expression of 9-84 for the Beta function
we have, when R(x) > 0,
(1) = 2 a,
f = 0 0
This suggests consideration of the function
(2) (f)(t) = + + .
We now prove that the series
(3)
s = 0
is uniformly convergent in the interval provided that
R{x) be greater than the larger of the numbers 1, X + 2, where 1 is,
as usual, the abscissa of convergence of the factorial series. Let
E{x) = G.
Then the factorial series converges when x = g-2 on account of
the way in which C7 was determined, and consequently the 5th term
bends to zero when 5 oo . Thus
lim
(or - 1) (7 ... {(7 + 5-2)
so that
r I /a + 5 — 2
lim \as =0,
S-+CO
and hence, given e > 0, we can find n, such that
It follows then that
8 = n s=^n *
which proves the uniform convergence of (S).
We can therefore integrate term by term and we get from (1)
(4) Q(ir)=:
h
An integral of the above form is known as Laplace’s Integral.
290
FACTORIAL SERIES
[10-2
We tave thus proved that, if R{x) be sufficiently large, the sum
function of a series of inverse factorials can be expressed by Laplace's
integral in the form (4).
Conversely, if (j){t) be given in the form (2) and if, for R{x) suffi-
ciently great, the series (3) be uniformly convergent for 0 ^ ^ L
the corresponding Laplace’s integral (4) can be represented by an
inverse factorial series.
The function may be called the generating function of the
factorial series.
When the function 0 (x) is given, the generating function (f> (t) is
obtained by solving the integral equation (4). It is easily verified *
that the solution is
where I is any number greater than the abscissa of absolute con-
vergence of the factorial series for (x).
Example 1. Find the sum function of the series
^lX{x+l) ...{x+s)'
Here = s~^, so that
Hi)
_l-< {l-<y
Thus
so that
1 2
+ '
--f- ... = - log t.
Q(x) =-rt^-iiogtdt=-~ = i,
Jo da? Jo x^
1 _ 0! ]J_ 2!
x(x+l)'^x(x-hl)(x+2)'^x(x+l)(x-i-2)(x+3)
The ratio of the (5 + l)th term to the 5th is
aj+l + s
so that by Weierstrass' criterion, 9*8, the series converges absolutely
if R(x) > 0. Hence yi = 0.
a ^0^® cq^^aplete discussion of the generating function, see Norland’s
jSenes a mferpolaUon, chap, vi.
10-2]
FACTORIAL SERIES
291
Again,
\ 11 1
^ as = 1-f ,^ + 5+ ... +- ' y-f log^^,
hence, from Theorem V,
X = limsup log(YH-logn) /logn = 0,
so that in this case X = pi = 0.
Example 2. Expand {x-a)-^ in a series of inverse factorials.
From (5), the generating function is
■j n+ico f-z
2m Ji^i^z-a
if I be sufficiently great. This can also be inferred Irom (4), since
1
=z
'-a Jo
By the binomial theorem,
= [!-(!
Hence, from (1),
a-fs-l
(1-0^
— -=-+1:
X- a X
^ ^ — 1
B{x, 5+1).
Thus we have Waring’s formula, namely
1 _ 1 a a(a+l) a(a+ l)(a + 2)
x~a~~ x~^ x{x-\-l) x{x+l){x^2)'^ x{x+l){x+2){x+2>)^ '
The formula can also be obtained from 3*1 (4), applied to the
function x~'^^ by interchanging a and x and putting p = - a.
By Weierstrass’ criterion, the abscissa of absolute convergence is
given by
fi. = E{a),
and since {x—a)~^ has a pole at x = a,
X = Ria).
Example 3. The series
1 « I a(a + l) a(a+l)(a+2)_
x{x+iy x{x^l){x+2) x{x+l){x+‘i){x+Zy
292
FACTOBIAL SERIES
[10*2
has the same abscissa of absolute convergence as the series for
so that
[JL = R{a).
On the other hand/by Weierstrass’ criterion, the series converges
conditionally if
0 < R{x-a-hl) < 1,
so that
X = 1).
These examples illustrate the result of 10*06.
10'22. Order of Singularity and the Convergence
Abscissa. If
f{z) ,
the order h of f(z) on its circle of convergence whose radius is
taken to be unity is by Hadamard’s dej&nition *
A = l + limsup^^i^.
^^00 logn
Multiply the generating function ^(0 in 10*2 (2) by
and we obtain
Ht)
t
00
(aQ + a^-f ^2 + ... + as) (1 - ty,
so that <^{t) It is holomorphic inside the circle | 1 - i | = 1. Thus,
if X>0, we have from Theorem V that the order of <f>(t)jt on
the circle | i - 1 1 = 1 is X-f- 1.
If X<0, the series converges; thus, if i5->0 along the
radius joining 1 to 0,
and hence
lim = ^( + 0),
i~->0
t (®s+l + ®s+2 + • • •) (1 - *)*>
and hence, again by Theorem V, the order of -i>( + 0)]/t on
the circle | Ij = 1 is X+ 1.
* P. Dienes, The Taylor Series (1931), p. 493.
FACTOBIAL SERIES
293
10-3]
10'3. The Transformation (x, x + m). Consider
QO I
a, 5!
= S
dt,
Jo
x(xi~l) ... (a; + 5)
where (f>{t) is the generating function 10-2 (2).
We have identically
Q (x) = \ ^
Jo
r- W = [1 ~ (1 - 2 a,(l -- ty = 2 6,(1 ty,
+
/m+1
V 2
+ . . . +
'm + 5- 1
so that
3^ y M
^ ^ (^ + m)(a;+m+l) ... (iCH-m+-5) ’
which we call the transformation {x, x^m).
If denote the convergence abscissa of the transformed series
we can shew, from considerations of order (10-22), that
X^ X, if R{m) >0, X > 0,
X~jB(m), if R{m) <0, X^ 0,
while in general
X^ > 0, if X < 0.
The case m = 1 is particularly simple, for then
= ag+ag_i + ... + aoj
so that
^ a;(£c+ 1) ... (a;+5) ^ {x+l){x-\-2) ... (cc+5+1) '
with Xi < X if X > 0, Xi := 0 if X < 0.
The transformation can sometimes be effected directly; for
example, using 10*2, Ex. 2, we have
1 1 ^ _l_ 'Kyi’\'Ci
x-a~^ (x-^-m)- (m+a) ~~ x-^-m^ {x-\-m){x-]-m + l)
(m+a)(m+a+l) ^
{x^m){x+m + l){x+m+2)
294
FACTORIAL SERIES
[10-3
It may be observed that if < a, the transformation {x, x+m)
gives the analytic continuation of Q {x) beyond the original half-
plane of convergence.
10-32. The Transformation (x, x/to). Starting from
Q,{x)==[
J 0
we make the change of variable = z, > 1. Then
1 ri 1
Q{x) = ^\ z- cf>{z'^)dz.
6) Jo
If 0) be large enough, j){z'^) will be holomorphic inside and on the
circle | ^ - 1 1 = 1, except perhaps at ;s = 0. Now,
1 <» 1
= 2] a,(l-2=)^
s=0
SO that
and therefore
fs,l{^) Ui + fs, 2 (w) ^2 "t • • • "t/s, « (®) ®5-
It follows that
n{x) = ^
b.s\ i
^0 a:(a:-|-co) ... (a: + sco) ’
which we call the transformation * (x, xj a).
If X > 0, considerations of order at the point z = 0 shew that
the series converges if R{x) > X(<o) where X(co) < X.
* See X<)rliuid, Series d,’ interpolation, chap. vi. For recent research on the
analytic continuation of factorial series of both kinds, see H. K. Hughes
iuIS ^ new results are
104] FACTORIAL SERIES 295
10*4. Addition and Multiplication of Inverse Factorial
Series. Suppose that we have the two series
a:(a;+l) ... (a;+s)
R{x) > X,
^2^(x) —
s!6.
E(x)>X'.
8=0 !K(a7+l) ••• (a:+s) '
If I denote the greater of the numbers X, X', we have evidently
which solves the problem of addition and subtraction.
The problem of multiplication is more difSicult on account of the
complexity of the coefficients in the product. The solution of the
problem is given by the following theorem due to N. Nielsen.*
The product of Q^(x) is developable in a factorial series of
the same form, convergent for i?(ir)>0, R(x)>l. These con-
ditions are always sufficient and generally necessary. The product is
x(x+l) ... (x-^-sY
where
=£(»»- 5) ! 5 ! K-s c„-8, , a. 8 = i: •
s=0 p=0 ^ V ^
To obtain a practical method of forming the product we use
Laplace’s Integral. Let
Q(a;) = f 9 (a) cZa, = f il'CP)
.1 0 0
Then
Q (x) f2jL (a?) = j* f (ap)*~^ 9 (cx) ij; (P) cZa d^.
JQJQ
Making the change of variable ap = we have
12 {x) {x) = [ X (0
•lo
where
* Rendiconti della R, Acc. dei Lincei (5), 13 (1904).
296 FACTORIAL SERIES [10-4
This result will determine the form of the product. The conver-
gence abscissa must be determined separately.
For example, consider
J_^i, , y(y+i) _ +
x-p X l)(ic+2)
1 1(9 ±ll ,
x-q X £c(x-f-l) l)(a;-|- 2)
JO
Jo
Here
den ~ .
p-q
To obtain the coefficients we can expand x(^) in powers of (1 - 1),
but it is simpler here to write
l_ _
x-p p-q
, P + q+l p^+pq + q^+3{p + q) + 2
x{x^ + 1)(cc-f2) ir(a;+ 1) (a;H-2j(a: + 3)
which is obtained by subtracting the second given series from the
first and dividing by p - q, and in fact this result could have been
obtained direct, without calculation. We infer that
1 2p -f- 1 4“ Sp + 2
{x-pf ~~ (r(a;-i-l)'^a;(2;+l)(a;-f-2) a;(cr+ l)(a; + 2)(a;-h3)'^
which agrees with the result of differentiating the first of the given
series with respect to p.
Actually, if = g, we have
^ (i) = f a“^ doi = log t
which gives the same series as before for {x-p)~^.
10-42] FACTORIAL SERIES 297
10*42. Differentiation of Inverse Factorial Series. If
we have
Q'(x) =: f t^'^^\ogt(f>{t)dt,
which is again developable in a factorial series.
In fact
so that
f2'(a;) =~
2
t=rl
Cti(\ Ct't Q/o i\ I
-9 + — ^ + ... + -4^)5!
5 ^-1 1 /
x{x-\-l){x^2) ... (07+5)
If X > 0, the order of (t) log t on the circle | i -- 1 1 = 1 is the
same as the order of that is to say, X+1, so that the series
for iT{x) has the same convergence abscissa as Q>{x), namely, X.
If, however, X < 0, we know that a? = 0 is in general a simple
pole of Q{x), and consequently a double pole of O' (a;), so that O'(rr)
cannot have a convergence abscissa < 0.
Hence, if X < 0, the convergence abscissa for Q.'{x) is in general
zero.
Example.
-1_ = 2 a(a+l) (a + s-l) ^ ^
X-CL X{X + 1) ...(X'^rS) Jo
The coefficient of (1 - in log^ is equal to the coefficient of
y® in (1 " y)-‘^ log (1 “ y), that is, in
which is equal to
d a(a + l) ... (a + 5- 1)
9a " s !
298
FACTORIAL SERIES
[10*42
Thus
1 ^ A a(oc + l) ...(a + g-l)/l - 1
{x-dY g^i ic(a?+l) ... (a?4-5) Va a+1 *“ a-fs-l
It will he seen that the direct application of the general formula
for Q'{x) leads to an equivalent but more complicated form for
this result.
The convergence abscissa is i2(a) whatever a may be, in fact
ic == 0 is not a pole unless a = 0, so that this is a case of exception
to the rule that the convergence abscissa is zero if X < 0.
Since oc(a+l) ... (a+s~l) = 5), the series for
can also be obtained by direct differentiation with respect to a.
10-43. An Asymptotic Formula. We have
(1) R{x)>0.
X Jo
Differentiate m times with respect to x and we obtain
Now by 6*4:3 (1), writing ~ 1 + ^ for t, we have
log-) =m(l-i) Zj — r-^ — •
y^o \f\ m+v
Multiply by and integrate term by term, then
r(w-fl) ^ ^ r(a;)r(m+v + l)
v=o (m4-v)v! r(a;-fm+v + l)
Multiply both sides by
r(a;+m+l)
r(a;)r(m+l)’
and we obtain, on writing s for v,
1)
(-Vf £.(.«
^io(2!+>w+l)(a:+m+2) ... {x+m+s)
, B(x)>0.
10-43]
FACTORIAL SERIES
299
From this we can infer that the development of the reciprocal of
the left side is given by a relation of the form
r(a;4-m+l)~" ... (25+5) ~ 0(^)5 say.
Differentiating with respect to m, we have
xrn+i^ T 4. (log x) (1 + Oq (a:) )
of
— _ o / ^
^ aj(a;+l)(x+5) ~~
Thus
Proceeding in this way we can obtain a relation of the form
r(^)
dm® r(aj+m+l)
= Q,(a;) + fi,_i(a;)log^+Q,_2(a;)(log^) +... + (l + fio(a:)) (log^
where Qo{x), ... represent factorial series which vanish when
a; = 4- 00.
It follows that, when R{x) is large and positive, we can replace
the right-hand side by its greatest term, namely (log“) •
We have, therefore, the important asymptotic relation that, for
large positive values of R{x)j
d® T{x) ly
dm® r(ir-f-m-fl) '^xJ \
which is useful in the theory of difference equations.
10*44. Integration of Inverse Factorial Series. Let
be a point interior to the half-plane of convergence. Then
mzmdt.
logi
300 FACTORIAL SERIES [10-44
The second term on the right is independent of a; (= C say), and
integrating 10*43 (1) with respect to x, we have
and therefore
f Q(x)dx-^{l)logx-G = f
J Xq J 0 ^
Now,
f il) = + flg (1 - t) + fflg ( 1 - f)2 + . . . ,
and from 643 (7), in terms of Bernoulli’s numbers, we have
Thus
log«
— 1 + 1- (1 — t) +
^2 v-1 v!
MziW
logt
“®1+(“®2 + 4®i) (1-0
/ \
Hence we have
f Q (x) dx = C+a^ logic 4- 2 . : ,
where
6q — - bi= - ^2+
6,= -<..„+i.,+ S^(-l)^’5go
This result is valid for R (x) >0, R (x) > X, unless Gq = 0, in
which case it holds for jB(a;) > X.
10-5. Finite Difference and Sum of Factorial Series.
The operations which we have hitherto considered, namely, multi-
plication, differentiation, and integration, are operations which are
simpler in their application to power series than to factorial series.
On the other hand, factorial series of both kinds are admirably
adapted to the performance of the operations ^ and its inverse.
10-53
FACTOKIAL SEEIES
301
Tims we have from
^ ” S, i(t!+l)...(x+l) '
(-l)Ai’W=
« = 0 ' o
(-l)”A^(^)= Ej-^r^s^r.{ /)■
Again, the equation
A w (a?) = 0 (ic)
clearly admits the general solution
“(“=) = " w (*> - ,s i(iTifrW+7) ■
where m(x) is an arbitrary periodic function of period unity, and
the principal solution is obtained by replacing zu (x) by a constant,
so that
where C is a constant whose value depends on the lower limit of the
summation.* Similarly, we have
^ Fit) At = C,-J:(- ly a,
It is clear that the operations A and ^ do not affect the conver-
gence abscissa since the coefficients are merely displaced, and the
limits of Theorem V are unaltered.
*It is convenient to indicate “indefinite summation*’ (the analogue of
indefinite integration) by omitting the lower limit.
302
FACTORIAL SERIES
[10-5
Example,
x-a ~~ sii) x{x+l) (x + s)
Thus
X
= ^(x)-
s
s = 0
1 a (a + 1) ... (a + 1?)
5+1 x{x+ 1) ... (x + s) '
the constant being zero, since both sides must agree when a = 0.
10*6. Newton’s Series. The series
F{X)=
s = 0 ^ ^
to which we shall refer as Newton’s series, converges in a half-plane
(10*04) limited on the left by the line R (x) = X, and converges
absolutely in a half-plane (10*06) limited on the left by the line
R(x) = [L, where 0 - X ^ 1.
When a; is a positive integer, the series reduces to a polynomial and
may therefore be said to converge at those of the integral points
which may lie outside the half-plane of convergence, but diverges in
a neighbourhood of such points. We shall therefore not include in
the region of convergence those integral points which lie outside the
half-plane of convergence. The convergence abscissa X is given by
Theorem V.
10*61. Uniform Convergence of Newton’s Series. We
now prove the following theorem due to Norlund : *
Theorem VIII, If Newton^s series converge at the point Xq,
where Xq is not a positive integer, the series converges uniformly at
every point of the sector S, vertex at Xq, such that
-^TT+T] < arg(cc-a;o) < Itc-t],
where 73 is positive and arbitrarily small, and R is any positive number,
* Bties dHnterpolaUm, p. 100.
10-61]
FACTORIAL SERIES
303
Let the given convergent series be
and let
__ (cc-l)(ic-2) ...(a:;-s)
Fig. 14.
We have to prove that, if x be any point of S (see Fig. 14), S
converges uniformly.
It is clear from the figure that we can find an integer n, such that
|arg(5-a;o)| <|7), if
and also that the line joining Xq to s subtends at any point of S an
angle ^ greater than i(7r+7]). The projection of this line on
the line joining s to x cannot exceed the length of the line joining
5 to cCq ; hence, if r = \x-Xq\,
\s-Xq\'^ |5~a;|+r cos (tt — (5i)> |5-a;|-fr sin \i] ,
so that
\s-x \ < 1 5 I - r sin Jt] ,
and hence
n\ rsin|-73 cosarg(5-a;Q)
iCn - s ' s-Xq \ cos arg (5 - Xq)
< I _ g-gp-IrsinT]
? — On $ Gq
FACTOEIAL SERIES
[10-61
304
where (Tq = -^ (^o)> s > Also we have
(2)
Now,
s-a!o+l '^s-Oo + l’
« _« {x-n)...{x-s)
” '^{a:o-n)...(a;o-s)’
and by (1),
{x-n)...{x-s) ^ (w-gQ-^sinv]) ... (s- gp-lr sin v]) _ y
{x^-n)...{x^-s) (w-go) ...(s-go) ‘
say.
Also, when x lies in S, u„_i is clearly bounded since n is fixed.
Suppose that | u„_i \<.K. We have then
\v,\<Ky„ F,<1,
so that the v, are uniformly bounded in 8.
Again,
X-Xa
Hence, using (2),
1 ““ j ^ ^
^ 2Z
5 — <Tft+l sinri
{Vs-V,
s+1^
This inequality is of exactly the same type as 10*11 (4), and the
required uniform convergence follows by precisely the same steps as
in that section.
It follows from this theorem that the sum function F{x) of
Newton’s series is an analytic function of £C in every closed region
which, together with its boundaries, lies in the half-plane of con-
vergence, for any such region can be enclosed within a sector of
the type given in the theorem.
1 0 • 63 . Null Seri es . Consider the binomial series *
(1 +«)-!= i:a*(®“^).
s-O \ S /
If 1 a I < 1, the series converges everywhere, so that X = - oo ♦
* K. Kjiopp, Infinite Series, p. 426,
10*63]
FACTOBIAL SERIES
305
If 1 (X I > 1, the series diverges everywhere (except at the positive
integer points), so that X = + oo .
If |oc| = l, the series diverges if li(cc)^0, converges
simply if 0 < jB(a;) 1, and converges absolutely if 5(£c) > 1, so
that X = 0, [i, = L
If a = - 1, we have
*s=0 ' ^ n~>*oo s~0
'x-V
S
Now,
2 ^ ) = coefiBicient of in +
« =0 ^ '
= coefficient of in (1 -
Thus
I' /x-l\ _ (n-cc) ... (2-a;) _ r(^~a;+l)
s J ' (n-l)! '^r(nyr(2-x) r(2~a;)
for large values of n (see 9-61). Thus when n- - oo , the right-hand
member ->0 if J?(2:) > 1, and -> oo if i?(a:) < 1. Hence the series
= Z -1)'
« = 0 ^
converges in the half-plane 2i(x) > 1, and is equal to zero for all
values of x in this half-plane. To such a Newton's series we give the
name null series. We note that when a = 1, ^(1) = 1.
10* 64. Unique Development. If a given function /(a;) be
holomorphic in the half-plane R{x) > I, and if this function be
capable of expansion in a Newton’s series which is convergent in the
half-plane R{x) > X, we can shew that the expansion is unique,
provided that I X < 1.
For, let the Newton’s series be
t=o '■ s /
Then
A^-a) = (-!)*«,
i'ACTOBIAL SERIES
[10-64
306
But ^(1) = f{l), since l^Kl, and hence
F{x)=
s^O ^ ^
and tte expansion is uniquely determined.
It follows from this theorem that no null series can have a con-
vergence abscissa which is less than unity.
10*65. Expansion in Newton’s Series. Suppose that we
are given a function f{x) which is holomorphic in the half-plane
R{x) > I, and that this function is representable by a Newton’s
series whose convergence abscissa is X, where n:^X<n-hl,
n being a positive integer, and Z X. Let F (x) be the sum function
of this series. Then
(1) Fix) = 2 ( - (''~s^) = i: A ^’(1) ^
as in the preceding section.
From 2-5 (2), we have
(2) ^F{1) ^ F{s+1)-{\)f{s) + (1) F{s~l)- ... + i-l)sF{l).
Now, if s ^ we can write this in the, form
(3) A^’(l)= E(-1)*-'(!)^(v+1)+S (-l)-'(")/(v-fl)
v==0 v—n
since by hjrpothesis /(v+1) = J'(v4-1) when v > n.
If we substitute these expressions in (1), we obtain for the coeffi-
cient of J'(v-t- 1), V ^ w— 1, the series
where ^{x) denotes the null series of 10-63.
The series ^ (a; — v) is likewise a null series whose convergence
abscissa is v-H 1, so that the contribution to (1) of the numbers .f (1)
Fi2), ... , Fin) is
10-651
FACTOBIAL SEBIES
307
wMch is also a mill series whose convergence abscissa is n. Thus it
appears that the sum function F{x) of the series (1) is independent
of the values of the sum of the series at the points 1, 2, 3, , n,
and consequently that we can arbitrarily assign the values
jF(1), F{2), ... , F{n) without altering the value of the sum-function
in the half-plane of convergence of the series.
Thus, if X > 1, the expansion in Newton’s series of a function
f{x) which can be so expanded is not unique but admits of infinite
variety.
In some measure to restrict the choice of series, we define a
reduced series as follows :
Let m be the smallest positive integer, such that the given function
f{x) is holomorphic for R{x) > m and continuous on the right at
X = m, so that
f(m) = lim/(w+s),
€—>0
where s 0 through positive values. A Newton’s series is said to
be reduced if the sum function F(x) be such that
F{x) — f(x), a; = m, m + 1, m + 2, ... .
If the convergence abscissa X of a reduced series be greater than
the integer m, and if we add to this series a null series whose con-
vergence abscissa is less than X, the new series has the same
convergence abscissa as the original series. A series obtained in
this way may also be called reduced. In any case the convergence
abscissa of a reduced series is the least possible, that is to say, no
other Newton’s series which represents the function can have a
smaller convergence abscissa.
Example. Expand (a;-a)“^ in a Newton’s series. We have
so that from 2-11 (7),
= (- 1)® 5! (aj-a-
Thus
(x-oc) ... (x-a-{-5) ‘
{x-~ 1) ... {x-s)
X-OL
308
FACTORIAL SERIES
[10-66
TKe ratio of the (54- l)th term to the 5th is
s-x
5 + 1-a
so that by Weierstrass’ criterion the series is absolutely convergent
if R{x) > B{cc). Also the series diverges if R{x) < i?(a), since
(a7“a)~^ is infinite at a; = a, while the sum of a Newton’s. series is
holomorphic at every point of the half-plane of convergence. Thus
X = |x = i?(a).
This series can also be deduced from 10-2, Ex. 2, by writing 1 - a
for a? and 1-ajfor a.
If we differentiate with respect to a, we obtain
1 ^ A r 1 1 . I n
(x-ccf ^(a-1) ... La-1 a-2
Proceeding in this way we can obtain a Newton’s series for
and so any rational function can be expanded in a
Newton’s series.
The above method fails if a = n, a positive integer. To obtain
a reduced series, let us take the sum-function to have the values
1 2-w’ ’ - 1
at the points a; = 1,2, ..., w — 1, and let us choose the value of the
sum-function at tr = ?^, s,o that the coefficient is zero. We then
have
x-n /ro(w-l)...(«-s-l)
+ S
s!
'(-1)*
where, from (3),
n-2
y*0
1
n- v-1
and putting a; = n in the series, we get
FACTOBIAL SEEIES
309
Tims ( - 1)" a, is equal to the coefhcient of in
■ (1 + «)» log (1 + 0 + ( - 1)” ^ j) (1 + ty log (1 + <).
and hence the coefficient of is
3v \v
where
^(1 + ^)" = (l+^)Mog(l4-i),
fficient of is
I; (v^ = - +s-37tt)
Also
log(l+i)
1 + ^
Hence
(-l)”o.=
Ss= l+4-+i+...+“.
. 1 ) [ ““ + ^s-n+1 + ^n-l + ^5 ~ ^n-l]
Hence finally we have the reduced series
=-s
(x- 1) ...
x-n /to(^-l) ... (n- 5-1)
(n-l)!(s-n+l)!
of which the convergence abscissa is n.
1 + J+ ... +
5-n+lJ’
10*67. Abscissa of Convergence of Newton’s Series.
Let the function f(x) be represented by the Newton’s series whose
sum-function is F(x). The method of the preceding section enables
us to obtain another form for the convergence abscissa.
If w be a positive integer such that 0 ^ ^ ^ X, we have
s^O \ S '
{-iya, = KF{l).
310
FACTOBIAL SERIES
10-67]
If we denote by 1(0) an arbitrary constant, wbicb we introduce
in front of tie sequence
F(l), F(2), F(3), ... , F{n)J(n+l),fin+2}, ... ,
we have
(1) S (-i)*AJ’(1) = sVi)‘[ A-f(0)+ ‘af(0)]
= F{0)+{-1)-^AF(0),
and bence, from Tbeorem V,
(2) X = limsup log I A^(0) I jlogn.
This formula for X is still valid when X < 0, if we remember that
J(0), i’(l), ^'(2), ... are now the values of /(O), /(I), /(2), ... .
We have in fact
^(0) =/(0) =S
fi=0
so that, from (1),
E a, = (-!)« Am
s=n
and (2) therefore still yields the convergence abscissa.
10*7. Majorant Properties. We give here some theorems
which indicate the nature of analytic functions which can be
expanded in Newton’s series. The proofs are lengthy and are
omitted. They will be found in Norlund’s Series F interpolation,
Chapter V. The first of these theorems is due to F. Carlson.
Theorem. Let F{x) be a function which can be expanded in a
Newton’s series of convergence abscissa X. Let a be a real nuniber
greater than Xandletx~a. = r where - < 6 < Then
where
|.X+l+<(r)
(1+r COS 6)^’
4'(6) — cos 0 log(2 cos 6) + 6 sin 0
Grind e(f) ieinds uniformly to zero as r oo ,
311
10-7] FACTORIAL SERIES
A sufficient condition for the convergence of the Newton series
of a function is contained in the following theorem due to Norlnnd.
We use the same notations as before.
Theorem . Let F{x) be an analytic function which is holomorphic
in the half -plane R{x)'^ a and satisfying in this half -plane the in-
equality
I I < (1 + - -Ire < 6 < I-tc.
The function F{x) can be expanded in a Newton^ s series whose con-
vergence abscissa does not exceed the greater of the numbers a, p 4*
For the more general series,
(x-c^){x-2c^) ... {x-S(o)
= 0 s\
Norland has proved the following :
Theorem. In order that a function F(x) should admit a develop-
ment of the above form f it is necessary and sufficient that F {x) should be
holomorphic in a certain half -plane R(x) > a and should there satisfy
the inequality
where C and k are fixed positive numbers.
It is here sufficient to take
ko^ < log 2 ^(0).
Any function which can be developed in a Newton’s series admits,
a fortiori, a development of the above form where 0 < co < 1.
10*8. Euler’s Transformation of Series. Let
be a power series in which converges outside the circle | i [ = JS,
If we write t = l + u and expand each term in negative powers
of u, we obtain
(-1)- (!)«—>
8 = 0
Since the original series converges if | { | > the second series
will certainly converge if | m | > 1 + i?, that is to say, the power
312
FACTORIAL SERIES
[10-8
series in u~'^ converges outside tlie circle whose centre is the point
t=:l and whose radius is l + JK. But by Weierstrass' theorem
on double series * we can interchange the orders of summation.
Hence we obtain
^(0= Sm— IS (-l)»-'(")j’(v).
Now, from 2-5 (2), we have
S(-i)‘-'(O^M = AJ?’(0).
y;=:0 ^
Thus we have Euler’s transformation, namely,
V -- V
The series on the right certainly converges outside the circle
I ^-1 j = 14.5^ but the series may also converge at points within
this circle.
10*82. Generating Function. Consider the Newton’s series
(1) F{x)=±iF{l)(^-'^).
«=0 \ 5 /
The function ^ (t) defined by
(2) m=±^
8=0 f
is called the generating function of the series.
To obtain the region of convergence of the series which defines
the generating function we have, by Carlson’s theorem, 10*7,
\F{s)\< e^iog2^\+, ^
and hence
'V \t\
Since
logg
lim e * = = 1,
* K. Knopp, Infinite Series, p. 430.
313
10-82] FACTORIAL SERIES
we see that the series for converges if
l<l >2-
Applying Euler’s transformation to we obtain
m _ V A-F(o)
where, as in 10-67, F{0) is to be replaced by an arbitrary number
if X > 0.
Multiplying both sides of (3) by « E(i- 1) + 1, we have
^(0=1:
«=o
AiXO) , V
(t-iy ^0
AF{0)
(«-l)»+l
(4)
since
Now,
— F{0\ 1 A-P(l)
S + 1
Ai^(0) + A^(0) =
r(x)
r(x-5)r(5+i) ’
and by the complementary argument theorem
TC _ ( “
sin7i:(a:;-s) ~~ sinjca; '
r(a;--5) r(l
It follows that
/x~ 1\ _ sinrar(a;) r(5 + l -cc)
\ s J~ (~l)^Tc ~r(5 + l)
and hence, if R{x) = a, we have from 9*61,
imy-')
= \C{x)AF{l)s-^(l+8)\,
where G(x) is independent of s and S -s-O when s -i» oo .
Since the series (1) converges, the absolute value of the sth term
tends to zero when s -> oo . Taking a = X+ e (e > 0), we see that
for sufficiently large values of s
A-E(l)| <s>‘+'.
314
FACTOKIAL SERIES
[10-82
Hence
A^(i)
\~t-i
so that the series (4) converges at all points exterior to the circle
ji~l| z= 1, and therefore the series (3) converges in the same
domain. From 10-22 and 10*67 (2), we see that the order of ^(t) jt
on the circle | 1 1 = 1 is equal to X+ 1.
10*83. LapIace^s Integral. We have, from 9*88,
(1)
r(^) ^ 1
r(a;“8) r(5 + l) 27zi
the path of integration being a loop which starts from the origin,
makes a positive circuit round i = 1, and returns to the origin.
Also, the generating function is, from 10-82 (4),
If we take R{x)>0, R{x) >7^+1 and multiply by we can
integrate term by term, provided that the contour of (1) be enlarged
into a loop I which starts from the origin and encircles the circle
I i-~l I = 1 and then returns to the origin in such a way that no
branch of I is tangent to this circle at ^ = 0. We thus obtain
Conversely, every integral of this form, where <f>{t) is holomorphic
outside the circle | i — 1 1 = 1 and is of finite order on the circle, gives
rise to a Newton’s series.
Example. F{x) ■- a®
Here
'V' -f
s=Q
= H-a
10-83]
and hence
FAOTOBIAL SEEIES
315
F{x) =
(«-l)»+v
dt
= a 2 C®-!)*
«=0
X- 1
\ $
If I a - 1 1 =: 1, a ^ 0, we have
m 1
t ~{t-l)-{a-iy
so that t=: a is a simple pole on the circle | ^ - 1 1 = 1 and the
order of <j>it) Jt is unity. Thus X = 0. If | a - 1 1 < 1, / i5 is
hoiomorphic on the circle and therefore X = - cx) .
The expansion can also be obtained directly from the identity
= a(l + a-l)^'‘^
10‘85. Expansion of the Psi Function in Newton's
Series. We have
Thus, from 10-82 (4), the generating function for 'I' (1) - ■^'(a:) is
<f>{t)=X
1
1
Hence
(-1)* 1
dt
8^1 S
Since ^ (1) = - y, we have
with convergence abscissa zero.
316
FAOTOBIAL SERIES
[10-9
10-9. Application to the Hypergeometric Function.
From 9-8, 9-82, we have
r(c)r(;c+c-6-l)
r(c-6)r(a:+c-l)
= F{l-x, b; c; 1)
b , &(6 + l) /x-l\ b{b + l){b + 2) fx-l\
c 1 / c(c + l)V 2 / c(c+l)(c + 2)\ 3
which is a Newton’s series.
The function on the left is meromorphic in the whole complex
plane, with simple poles at 6-c+l, b-c, b-c-l, The con-
vergence abscissa is therefore 6 - c-f 1.
Writing c = 6 + 1, 6 = 2^, we have a Newton’s series for the
reciprocal of the Beta function, namely.
1 _ 1 l_ (x- ^ L
y ^+1^ 1 / y+2 \ .2 / y+Z \ 3
with convergence abscissa zero.
If we put 2/ = ^+1, where n is a positive integer, we obtain a
Newton’s series for the inverse factorial, namely,
^ 1 1 I
a:(x+l) ... (£c + ^^) n+1 ?i+2 v 1 / n + 3 \ 2 /
and for w = 0,
i= 1-1 ('*-!') H.il'”-'')-
* 211 J*3\ 2 /
Since
we have, by summation,
"if (x) = constant+ X’ f ^ ^ .
g^0^5+l/ 5 + 1
Putting a; = 1, we have for the constant the value 'SE^(l) = - y,
whence we obtain once more the result of 10-85.
EX. X]
FACTORIAL SERIES
317
EXAMPLES X
1. Prove tkat
111!, 2!
a; a;+l'^(a;+l)(a:+2)'^(a;+l)(iE+2)(a:+3)
+ . . 3!
{x+ 1) {x+ 2) (a;+ 3) {x^A)
2. Prove that the series
^Qxlxi-l) ... (a?+5)
represents a meromorphic funetion in the whole plane. Transform
the series by [x, 5C+1) and shew that the resulting series
^o{x^\){x-\-2) ... (x+a-hl)
has the convergence abscissa zero. What is the explanation of this
result ?
3. Establish the transformation by starting from the
1
integi'al “ = 1 dt and its limte differences
-=:\
x{xi-l) .,,(X + S) Jq ^ '
with respect to x.
4. Shew that the derivate of the function
which is meromorphic in the whole plane, has convergence abscissa
zero.
5. Prove that
* ' ' />» /»*_L 1
X x+1 x-i-2
~ V
x(x+l) ... (a;+5) '
and that the factorial series represents a meromorphic function in
the whole plane. (See also Ex. 4, p. 267.)
318
FACTOEIAL SERIES
[EX. X
6. Prove that
, . 2s+l
s! sm —7; — Tc
•^qx{x+1) ... (a:+s)’
R{x) > 0.
7. Prove that
Pi(^) =
^■'■22(a;+l)'^2S(a;+2)'^"' 2la:(2;+l) ... (a:+s)’ '^(^) > 0-
8. Prove that
s\
2^-1
and that the series is absolutely convergent.
9. Prove that
s!6.
where
^a;(a;+l) ... {a:+s)
*-l
, R{x)>0,
h = y lzlL_
” M s . 3«-‘'+i ’
and that the series is absolutely convergent.
10. Prove that
s\K
, R{x)>0,
where
,^ix(x+l) ... (a;+s)
and that the series is absolutely convergent.
11. Prove that
l-(a:-l) Pj(a;- 1) = y + 5fa:)>l
^ ^ox{x+l)...{x+sy
and also that
l-(a!-l)Pi(ic-l)
^g![l-2+3-... + (-ma+l)1
.=0 (a;+l)(a:+2)...{a;+s+l) ’
R{x)>0,
so that the second series is convergent in the strip 0 <R{x) <1,
where the first series is divergent.
EX. X]
FACTORIAL SERIES
319
12. Obtain the transformation (k, a;+l) by taking F^{x) = - in
SO
the product F{x) F-^{x) of two factorial series.
13. Prove that
^{x+y)-^ {X) = ["
jQ i — e ^
where R{x) > 0, R(x+y) > 0.
14. Prove that
•(x+y)--p(;r)c= 1:1^41' (2/-^).
' s + 1 a:(a;+l)...(a; + s)
15. Prove that
— J_= Bt\^\x+s)
(a;-a)n+i ^^\nJ x{x+l) ...{x+s)
16. By integrating a:-i(a:+l)-i, shew that
log 1 +
fi-1 T>(s-1)
X' X 2a;(a:+l) ^^2 («- 1) a:(a:+ 1) ... (k + s)
= V (-l)-‘B<|:i>
{x -f- 1) (x H-2) ... (a:+s)
where J? (ir) > 0 in both cases,
17. By summation of both sides in Ex. 16, prove that
•!'(,) = iog*-5j+ .i'.Sirrif jjiTTTrcrTij
= i„g.-L
^ a: s (cc+l) ... (x+s)
18. Determine the abscissa of absolute convergence of the
S(-ir
fi=0
320
FACTOKIAL SEBIES
[ex. X
19. Bxpaad in a Newton’s series
(l+a)”+(a;-a)-i, !al = l, a=j=-l,
and determine the abscissa of absolute convergence.
20. If
»=0 r^O
shew that
r(«) = 2;(-i)M.(^*j), x>o
F(x) = F(0)+Z(-iyB,(^), X<0,
S = 1
and that in each case the abscissa of absolute convergence is X.
21. Shew that the results of applying p times in succession each
of the above transformations yield
s=0 \ 5 /
where
r=o o — r
/p + r-s- 1
\ r — s
22. If/(»)= (^_2) -i- j shew that the equation
f(n) = 1 has the single root m = \ x^\ -ir 2 R(x).
Hence prove that the greatest value of (“'“Ml for fixed x
\ n /
occurs when n = [m], where [m] denotes the greatest integer
which does not exceed m.
23, Prove that
FACTORIAL SERIES
321
EX. X]
24. Prove that
where {x) is Laguerre’s pol3nioinial defined by
25. Shew that the hypergeometric function F{l-x,b\ c \ a)
represents a Newton’s series which converges everywhere if ] a ] < 1,
and that the generating function is
CHAPTER XI
THE DIFFERENCE EQUATION OF THE FIRST ORDER
11-0. The Genesis of Difference Equations. Let ^(a;)
denote an arbitrary periodic function of x of period unity, so that
ti7(a:+ 1) = tEr(a:). From a relation of the form
(1) F(a;, u^, xu{x)) = 0,
we obtain, by performing the operation
(2) ■ F{x+1, n}{x)) = F(x, ■w{x)).
The elimination of ©(a;) between (1) and (2) leads to a relation
of the form
(^) 4>{^> M*+i) = 0,
which is a difference equation of the first order, of which (1) may be
regarded as a comvplete ‘primitive. Observing that
«a!+l = +
the equation (3) could also be exhibited in the form
W Mx, AMx) = 0.
The problem to be envisaged is then, given a difference equation
of the form (3) or (4), to obtain a complete primitive of the
form (1). That such a problem is capable of solution is by no
means obvious, nor, supposing solution to be possible, are we
entitled to expect a solution in compact form. The proper
attitude is rather to regard a difference equation as possibly
defining a class of functions and to study the properties of these
functions from the form of the equation. In Chapter VIII we
11-0] THE DIFFERENCE EQUATION OF FIRST ORDER 323
have established the existence of a definite function which satisfies
the simplest possible difference equation,
but even there we have seen that the problem is not entirely simple
and that must be suitably restricted. In the present chapter
we shall consider only equations of the first order, and it will
appear that, except in the case of the linear equation, very little
is known of the theory. We shall denote the independent variable
by X and the dependent variable by u^, u{x), or u, according to
convenience.
Example 1. Assuming as complete primitive
Uy, = mx-\-
form the corresponding difference equation.
We have
whence
Example 2. Assuming as complete primitive
:= tETj a® + ©2
form the corresponding difference equation.
We have
+ ttTg 6®+^,
whence, eliminating a®, have
^a; ^x+l '^a:+2
la a^ =0,
1 h 62
*^a:+2 ~ (a + 6) "t ^
or
2
A^^a;-'(<^ + ^"'2) A'?^a;"t{a6-a-6 + l)t^a; = 0.
Either of these forms is a linear difference equation of the second
order with constant coefficients.
324 THE DIFFERENCE EQUATION OF FIRST ORDER [Il-oi
11 -01. The Linear Difference Equation of the First
Order. The general form of the linear equation of the first order is
(1) a{x)u{x->rl) + h{x)u{x) = c{x),
where a{x), b{x) and c{x) are given functions of x.
If we can find a particular function it^lx) which satisfies this
equation, we have
(2) a(x)u^{x+l)+b(x) U]^(x) — c{x).
If we now put uix) = u^(x)+v(x) in (1), we obtain, by sub-
tracting (2),
(3) a(x)v{x+l) + b{x)v{x) = 0.
Thus the general solution of (1) can be regarded as the sum of a
particular solution of (1) and the general solution of the homogeneous
linear equation (3).
This point of view is convenient in that it applies to linear equa-
tions of any order, but later we shall see how, in the case of the
general equation of the first order, it is possible to proceed at once
to a symbohc solution.
11-1. The Homogeneous Linear Equation. The general
type of this equation is that of 11-01 (3). Dividing by a{x) and
changing the notation, this can be written in the form
(1) u{x+ 1) = p[x) u{x).
The general method of solving this equation is as follows ;
Taking logarithms
logM(a;-t-l)-logM(a:) = logp(a:).
Hence summing the function on the right, we obtain
X
logu{x) = ^logp(f)Af-Fro(a:),
where w{x) is an arbitrary periodic function of x of period 1.
Such an arbitrary periodic function we shall in future denote by
IM] THE DIFFERENCE EQUATION OF FIRST ORDER 325
w and call an arbitrary ''periodic,” the argument x being implied.
Thus we obtain
X
i{x) = exp(OT+ ^logp(0 AO
C
X
(2) = ®iexp(§log3)(i) a0>
where = exp(ttr) is an arbitrary periodic.
The arbitrary constant c does not of course add generality to the
solution. This constant c may therefore be given any particular
value which is convenient for the purpose of summation. The
solution obtained in this way exists in so far as log j) (x) is summable
in the sense described in Chapter VIII. Moreover, in view of the
possible many- valued nature of the right-hand side of (2) it may be
necessary to make suitable cuts in the x plane. The important
point to observe is that the general solution of the homogeneous
linear equation contains an arbitrary function of period unity,
which can therefore be defined in a perfectly arbitrary manner
in the strip 0 < R{x) < 1.
The general solution of (1) is therefore only analytic if be
analytic. Moreover, the solution of ( 1 ) is only completely determined
when the value of u (a?) in the strip 0-^R{x) <l is assigned. In the
case of a differential equation of the form ^ (x) j/, the solution
is determined when y is given for a particular value of a; : in the
corresponding case of a difference equation, a particular value of
X for which u (a?) is given does not in general determine the solution.
Consider the equation
(3) (a; + 1) = 6^® u (x).
Here
u{x) w exp ^ A ^ J •
c
0 =0,
u(x) = m exp (jBg (x) )
= m exp {x^ — x+^).
Taking
326 THE DIFFERENCE EQUATION OF FIRST ORDER [IM
Introducing Bernoulli’s function P2,{^), which has the period 1
and coincides with B^i^) in the strip 0 < i? (a;) < 1, we can write
(4) u{x) = e “ ■"
where is an arbitrary periodic.
Let us now seek that solution of (3) which is equal to cos 2tzx in
the strip 0^R{x) <1,
Clearly we have the solution required if = cos 27Uir, that is to
say,
u (x) = cos 27ZX Pi
This is an analytic solution valid for all values of x, and con-
tinuous at a; = 1.
Suppose, again, that we require the solution of (3) which shall be
equal to x in the interval 0 < cc < 1. Bernoulli’s function Pi(cc) is
equal toaJ-Jin 0<cc<l and has period unity. The required
solution is therefore
u{x) =
This solution is discontinuous at a: 3, - 2, ~ 1, 0, 1, 2, ... .
More generally, if u {x) is to reduce to a given function f{x) in the
interval 0 ^ cc < 1, we expand f{x) in a Fourier series in this range
and substitute this Fourier series for in (4).
The above illustrations should sufiiciently shew that the nature of
the problem of solution of a difference equation is very different from
the corresponding problem in differential equations.
We cannot, for example, obtain a definite answer to the problem of
finding a solution of (3) which reduces to a constant h when a; = 0,
for the arbitrary periodic is now subject only to the restriction
that axjL = ^ when a; = 0. If, however, the values of x with which
we are concerned be of the form cc = a + n, where n is an arbitrary
integer and a is a constant, the situation is entirely changed, for in
this case we are not concerned with values of x other than those
assigned, and the solution of (3) which reduces to the constant h
when rc = a is now perfectly definite, being in fact
u{x) =
This type of problem is of frequent occurrence in the practical
applications of difference equations, but it must be borne in mind
IM] THE DIFFERENCE EQUATION OF FIRST ORDER 327
that this definiteness of the solution can only be obtained under the
special circumstances mentioned.
From the form of (2) it is evident that if we have two particular
solutions %(a;) and u^ix) of (1), then u^{x) = mu^{x), where ttx is a
periodic (not arbitrary), and further, that if we are in possession of
any particular solution ^1(0;), the general solution is mu^{x) where m
is an arbitrary periodic. We shall now investigate various particular
methods of finding a particular solution of (1) which may in special
cases be more conveniently applied than the general method just
explained.
11*2, Solution by means of the Gamma Function.
Rational Coefficients. We have seen in Chapter IX that the
equation u{x+l) = xu(x) has the particular solution u(x) = r{x).
Now, consider the equation
(1) w(x-l-l) =-r.(a;) u{x),
where r(x) is a rational function.* We can therefore suppose
^(a;-ai) {x-ql^) ... {x- dj,)
r(x)
(x - pi) (x - ^2) • • • - (^z) ’
where neither the oc^ nor the are necessarily all distinct.
Since
r(a;+l -ocj.) ;= (a;-a,.)r{X”ad
and 0®+^ c . 0®,
it is evident that (1) has the particular solution
(^)
u {x) =
T{x-ol^)T(x-o^2) ... r(a;-afc)
r (a; — Pi) r (x — P2) • • • r “ Pz)
Since is an integral function, it follows that the particular
solution found in this way is meromorphic in the whole plane with
poles at the points - n,
j i — 1 , 2, . . . , jfc,
= 1,2,3... .
The general solution is obtained by multipljdng the above by an
arbitrary periodic.
328 THE BIFFEREHCE EQUATION OF FIRST ORDER
Example 1. u{x+l) ^ 2 (x-\-\) u (x) .
2(a?+l)
[11-2
Here
so that
*•(0:) =
/ \ o» r(a:+l) m2’‘x
u{x)-m2 p(3.)r(a;)~ r(a:) '
Example 2. The equation with linear coefficients
(ax+b) w (a;+ 1) + {cx+ d) u (x) = 0.
Divide by a and write
The equation then assumes the form
{x^-e)v{x^\)-{x+f)v{x).
Finally, writing x for ic+e, we reduce the problem to the solution
of the form
xw(x-{‘l) = {x-oi)w{x),
A particular solution is
w{x)
r(l)r(a?~Qc)
-V{l-x)T{x)
whence, from 10*9,
which is a Newton’s series convergent for (a; - a) > 0.
11*3. The Complete Linear Equation of the First Order.
The general form of ll-Ol reduces at once to
(1) u{x+l)-p{x)u{x) ^ q{x).
We have seen that when q{x) = 0, we have the special solution
(2)
v^{x) = expj^^ logJ>WA<]-
To obtain tie general solution of (1), put
«(») = Ml (*)«(»),
11-3] THE DIEPEKENCE EQUATION OF FIRST ORDER 329
and we have, from (1),
v^{x+l)v{x+l)-f{x)u^{x)v{x) = q{x).
Now, Uy{x+ 1) = f{x) Ui{x). Hence we have
tii(a!+l) An(a:) = g(a:),
so that
= " +
C
Thus the general solution of (1) is u{x) =
X
?(s) 1
~ a; “1
O
/O \
As
exp
c
l^exp^^ logpWA^jj
w
c
where w is an arbitrary periodic and c an arbitrary constant to
which any convenient particular value may be assigned.
Example. u{x+l)-e^^u(x) Zx^
Here we take, as in 11*1 (3),
ici{x) =
Putting u{x) = V (x) we obtain
-y (a?) = dx\
v{x) = m + B^{x)^
u{x) = txT 6-®2 (■'*') -I- jBg (x)
11*31. The Case of Constant Coefficients. The linear
equation of the j&rst order with constant coef&cients is
u{x-\-\)-\u{x) <f>{x),
where X is independent of x.
If <l> (x) = 0, we have
u{x) w X®,
so that we can take the special solution
= X®”^-
330 THE DIFFERENCE EQUATION OF FIRST ORDER
Putting u (as) = X®-'- v (x) , we have
Av(x) =
whence
I
[n-31
u(x)
c
An interesting particular case of this equation is
u(x-hl) + u(x) = 2
corresponding to X = - 1. The general solution is
t (aj) = ( - ziy-h f
L J c
t s^Q x + s
TJie equation has therefore, as a particular solution, the function
This function g{x) has interesting properties, some of which are
given in Examples IX (4).
11*32. Application of Ascending Continued Fractions,
ibiother method of obtaining a particular solution of the complete
equation is as follows. The general equation ITOl (3) can be written
in the form
a(x) u{x) = u(x + l)-]-b{x),
so that
^ ^ a{x)
and by continued application of this result we have for u{x) the
ascending continued fraction *
u{x) =
6(a;)4- •
6(rr+l)+-
b{x + 2) +
6 (32 + 3) + '
a{x+3)
a(x + 2)
a{x-^l)
a(x)
* L. M. Milne-Thomson, Proc. Edinburgh Math. 80c. (2) 3, 1933.
11-32] THE DIFFERElSrCE EQUATION OF FIRST ORDER 331
wMch is equivalent to the infinite series
+ ^(£+.1) . + b^. + 2)_
^ ^ a{x) a{x) a(x-^l) a{x)a{x~\-l)a{x-\-2)
The general solution is obtained by adding to this the general
solution of the homogeneous equation
a{x) u{x) ~ u{x-\-l).
As an application, consider
u{x-{-l)-xu(x) ~ -e-p
We have the particular solution
u(x)
e-P p® prp e^P p®"^^
X x(x+ i)'^ x{x-i- i)(a:; + 2)'^
and the general solution
“<*) =
The above particular solution is an Incomplete Gamma Function
(see the next section).
11 ‘33. The Incomplete Gamma Functions,
the last section that the equation
u{x+l)-x u{x} = - e -P p^'
has the particular solution
(1)
P{x; p) = c-^P*E;;.7
... (x-f.s*)'
We saw in
The factorial series converges in the whole plane, that is, X = - co ,
with the exception of the points 0, -1, -2, ... which are simple
poles, so that P{xi p) is a meromorphic function of x in the whole
plane.
The generating function of the series (10*2) is
Tlius
P(x', p) = e~i‘ p® j dt
• ®
(2) = I ^ (pi)®-^ e-o* p = f'’ <®-i e-« dt,
Jo Jo
the integral representation being valid only if R{x)> 0.
332 THE DIFEERENCiE EQUATION OF FIRST ORDER [11-33
If we expand e~* and integrate term by term, we obtain Mittag-
Leffler’s partial fraction development.
p)
^0 s!(a:-l-s)’
which is valid in the whole plane and puts in evidence the poles at
0,-1, -2,....
The residue at the pole x = -n is
(-1)"
n\ ’
which is the same as the residue of r(a3) at x-= -n (see 9-55).
Hence the function
Q{xi p) = r(a;)- P(a:; p)
has no singularities at a finite distance from the origin and is there-
fore an integral function.
Thus
Q(a! ; p) = I °° er* dt - T e-* dt
JQ Jo
(3) =
J 0
whicli is valid for all values of x.
Since
r(a;+l)-a;r{a?) = 0,
it follows that Q{x; p) satisfies the difference equation
u(x+l)-xu(x) = p®.
On account of the properties (2) and (3) P(x ; p), Q(x; p) are
known as Incomplete Gamma Functions. The special functions
which arise when p = 1 are called Prym’s Functions.*
11*34. Application of Prym's Functions. We can use
Prym’s functions to solve the difference equation
(^) u(x+l)-xu(x) = R(x),
where 2i(x) is a polynomial.
♦ F. E. Piym, J. /. retne u. angew. Math. 82 (1877).
n-34] THE DIEEERENCE EQUATION OF FIRST ORDER 333
Expressing R{x) in factorials by the method of 2-12 or by
Newton’s Interpolation formula, we have
(2) =
gasO \6/
where n is the degree of R{x),
Now, let
<=o \ « /
Then
fix+l)-xfix) =:±8\
where 6n = = 0,
If we choose so that
5 = 1, 2, 3, ... n
we have, since = 0,
n
b, = - 2 ®t. S= 0, 1, 2, n-1.
These equations determine f{x) completely, and if we now write
u{x) = v{x)+f{x),
we have, from (1),
n
(3) v{x-^l)-xv{x) = as = A,
say.
Now, from 11*33, Prym’s function P(x; 1) satisfies the equation
P(x4'l ; l)-xP(x; 1) =
and therefore (3) has the particular solution
v(x) = -eAP(x; 1)
and consequently the general solution
v(x) = mr(x)-eAP(x; 1),
so that the general solution of (1) is
u(x} =:f(x}+mr{sc)-’eAP(x; 1).
334 THE DIFFERENCE EQUATION OF FIRST ORDER [11*4
11*4. The Exact Difference Equation of the First
Order. Very little is known about the theory of difference
equations which are not linear. There is a fairly complete theory
of the linear equation, including the exact linear equation, which
will be considered in a later chapter. Here we propose to develop
the outline of a theory of the exact difference equation of the first
order, but not necessarily of the first degree.*
If we denote, as usual, the independent variable by x and the
dependent variable by u{x) or u, we can write
h = A^(^) = = u{x+ l)-t6(a;),
and the general difference equation of the first order is of the form
(f) {x, u, h) = 0, or <f> {x, u, ^u) ~ 0.
We shall use the symbols h and A according to convenience to
denote the same operation. We proceed to consider such equations
of the first order as can be put into the form
(1) M {x, u)-^N{x, u,h)h = 0,
or its equivalent
M (x, u) + N(x, u, == 0,
where M {x, u) is independent of A This is an equation of the
first order, but not necessarily of the first degree.
It should be observed that A ^ is not, in general, constant, so that
the dependent and independent variables are not interchangeable.
In this respect the problem is very different from that of the
corresponding differential equation.
D efi n i t i o n . The equation
M[x, u) + T( {x, % A"^) A'^ = ^
is said to be exact, when a function f{x, u) exists which is independent
■of A is such that
(2) M{x, u) + N(x, u, A^) A^ = Afi^y
where
Af{^, =f{x-hl, u(x-{-l))-f{x, u{x)).
. *L. M. MiMe-Tliomson, “The exact difference equation of the first order,”
Proc, Camh Phil. Soc., 29 (1933).
11-4] THE DIFFERENCE EQUATION OF FIRST ORDER 335
Since u{x-\- \) = u^h, we have, from 2*105 (2),
Af{x, u) = '^)+'h /S,uf{x+l, u).
where we regard h as unaffected by either of the partial operators
Ax) Au*
k
The condition that (2) should be exact is clearly satisfied if, and
only if,
M{x, u) = Axf(^, w),
N(x, u, h) = Auf{^+'^, y)-
Using 2’ 105 (3), we see that a necessary condition is, therefore,
(3) AxN{^^ % A) = AuM.{x-\-l, u).
h
We shall now shew that this condition is su£B.cient to ensure that
(4) M{x, u) + N{x, u, Aw)A^ =
but that an additional condition is required in order that /{x, u, h)
shall be independent of h.
We write
X
V{x, w) = § M{t, u)/S.t,
so that, from the definition of the sum.
and hence
Axy{^, = M(x, u),
An Ax F(a;+1, u) =: 'W) = AxN{x, u, A),
k h
from (3). Using 2*105 (3), we can write this in the form
Ax{N{x, U,h)-Auy{^+h'i^)} = 0,
h
N {x, u, A) = Am 1, u) + cy+ Am
h h
and thus
336 THE DIEEERENOE EQUATION OF FIRST ORDER [II-^
where the last term represents a function independent of x, and ro
is an arbitrary periodic in x. We have, therefore,
M (x, u)+N (a?, u,h)^u
= u) + h Att F(a;+1, u) + h At* F{u, h) + mh
h h
— A u) + F{u, 'h) + mu\.
naiTig 2*105 (2). This proves that (3) is sufficient to ensure (4).
Also, feom (4), we have
(5) Ax/(®, u,h) = M [x, u), A»/(a;+ l,u,h)=N {x, u, h),
h
and therefore, summing with respect to x^ regarding u as constant,
we have
f{x, u, h) =
where (f>{u, h) is independent of x, and zzTj is an arbitrary periodic.
If we write a;*h 1 for a? and then operate with Aus we have from (5)
h
a;+l
h) = N{x, u, A)- A« Q w) A^-
h h O
C
The left hand is independent of a?, and therefore the right hand
is independent of x, and h) can be obtained by summation
with respect to u. This introduces another arbitrary periodic
function of u of period h, that is to say, an arbitrary periodic in x
of period unity. Thus finally we have for /(a;, u, h) the expression
* tt a>+l
w+ g M(t, u) At+ g \n{x, V, ^)- A» g M{t,v)AtjAv,
c Cl c
where, in the summations with x in the upper limit, w, v are to be
regarded as constant.
That the above expression for f{x, u, h) gives
^ = -5f(a?, u)
11-4] THE DIPEEBElSrCE EQUATION OP FIRST ORDER 337
* is obvious. To verify the second formula of (5), we have
A
flj+l a;+2
= Au § M(i, u) u, ^)-Au§ M{t, u)At
c c
x+1
= N{x+1, u, ^ M{t, u)/^t
C
= N(x+1, u, h)- ^^M{x+l, u)
h
= N(x+1, u, h) - u, h) from (3),
= N(x, u, h).
We have thus proved the following :
Theorem. The necessary and sufficient conditions that the differ-
ence equation
M{x, u)->rN{x, u, Aw) Am = 0
should he exact are
(A) Au M{x+l,u) = Ax M. h),
h
(B) that
U Z-hl
h = Aw;
^ f^N{x,v, h)-/^^^M{t, ^))A<|A1’
should be independent of h.
If these conditions he satisfied, the primitive of the given equation is
X 11
^ Jkf (t, m) A f + ^ jiV {x, v,h)-/^^^M {t, ») A <1 A i’ = ro,
where m is an arbitrary periodic, and where in those summations
with X in the upper limit u and v are to be treated as constants.
The lower limits of the sumrmtions are arbitrary and may he
chosen to have any convenient values.
With regard to condition (B), I have not been able to obtain any
simpler formulation in the general case, but, when
F{u, h)=:N (x, u, h) - Au
h
«+i
ilf (^, w) A ^
e
338 THE DIFEERENCE EQUATION OF FIRST ORDER [114
is a polynomial, we can use Bernoulli’s polynomials, of order - ] ,
to simplify the process. As in 6-1 (3), we write
so that
ht
{u I h),
iu\h).
If then we can put F {u, h) in the form
m
F{u, =
^-=0
where the are independent of u and A, condition (B) will be
satisfied, since we then have
§
h i,=0 v-ri
v+l
where ifc is a constant. See also Ex. XI 30.
Example. Find the condition that the equation
a{^u)^-hbuj\u-{-c ^u+(l>{x) = 0
may be exact, where a, 6, c are constants.
Here
M{x^l) - <j>(x + l), N{x, u, h) ~ aJi + bu + c,
so that condition (A) is satisfied.
Condition (B) will be satisfied if we can find p and g, independent
of u and h, such that
Thus we must have a = b ^ q, c p, so that the equation
is exact if, and only if , 6 =+ 2a, in which case the primitive is
X u
that is,
X
cu+au^ = m.
11-41] THE DIFFERENCE EQUATION OF FIRST ORDER 339
11*41. Multipliers. When the given difference equation is
not exact, we are naturally led to consider the possible existence
of a multiplier which is the analogue of Euler’s integrating factor
for a differential equation.
D efi n i t i o n . Given the difference equation
M {x, u) -^N {x, u, H^u) 0,
[jL [x, u) is said to be a multiplier when the equation
[i{x, u)M{x^ u) + [i{x, u) N {x, u, l^u) ^u =. Q
is an exact equation.
For \l{x^ u) to be a multiplier, a necessary condition is that
[x [x, u) should satisfy the partial difference equation
-w) N{x, u, h)] = u)M{x+l, u)].
h
Any particular solution of this equation is a potential multiplier,
but in every case we must test whether condition (B) is satisfied.
This equation can be written
AJi^iV^Au ((xilf)] = Au[[xAf].
h h
11*42. Multipliers Independent of a?. If a multiplier in-
dependent of X exist, we must have
h h
which leads to
^ _h {x,u,h) + M{x+l,u)
\i(u) ” - M{x+l,u+~h) ■
A necessary condition for the existence of a multiplier independent
of X is therefore that Q{u) should be independent of x. If this be
the case, we have
Al0g|A(M) = Tl0gQ(M),
h
whence
u
[a(m) = exp ^log Q(i;) A ® •
This is a multiplier, provided that condition (B) be satisfied.
340 THE DEBTBBENCfE EQUATION OF FIBST OBDEB [11-42
Bmmph. (a:+l)A«(2+^Aw) + « = 0-
Here
Q{u) = (2k+^ + uj -r (u+h) =
Thus
|ji (u "f* fe) u-^h
fx(w) u ^
and we can take [x(w) = u, and tke equation becomes ^{xu^) = 0,
wbence xu^ =
11*43. Multipliers Independent of w. For a multiplier
independent of u we have, from 11*41,
Aa;[pj.H N {x, u, ii)] = ,p(a;+ 1) Au M (a;+ 1, u),
h
whence we obtain
(i(!C+l) _ _ N(x,u,h)
^t(x) - ^ W - N{x+l,u,h)-A„M{x+l, u)’
h
and T {x) must be independent of u. If this be the case we easily
obtain
X
= exp A .
which is a multiplier, provided that condition (B) be satisfied.
Example, (a;4-l)tz2(a3+l)~(ir-l)t^2(2,j ^
Since w(a;-f 1) = w(a;)+ A^(^^)j we have
(a;+l)(2'W+A^) Aw+2w^ = 0,
so that T {x) = (a?+ 1) / x, and hence
p(a?+l) _ a?+l
fi.(a;) “* a;
Taking {^(ic) = x, we have
x{x~l)u^{x) = w.
Alternatively
[!.(») = exp [ ^ {log(«+ 1) - log «} A <]
= expfloga;] = x.
11-5] THE DIFFERENCE EQUATION OF FIRST ORDER 341
11*6. The Independent Variable Absent. Haldane's
Method. The general equation of this type is
Aw) = 0,
where A'^ =
When this equation can be solved for A we have
(1) A^ = u{x-hl)-u{x) = <j)[u{x)].
An elegant method of obtaining a solution of equations of this
type has been devised by Haldane.* The method is as follows :
Taking the equation
(2) u{x+l)-u{x) = k(j>[u(x)],
let us try to satisfy it by assuming that
1 Cu(,x)
(3) x — j] w{Vfk)dv,
^ J u(fi)
where
(4) w{v,k) = j:^Mv)-
We have then, from (3) and (2),
ru(x+i) , » .
= w{v,h)dv=\ S /s W
Ju{x) Ju 8 = 1 *•
Cu+H ® Js-l
where u = u{x)y ^ = ^{u).
In order to obtain a recurrence relation for the functions we
assume that the series can be integrated term by term, which will
certainly be the case whenever the series is uniformly convergent.
Put
(5) =J /,(«>) dv.
Then
= F,{u+hf>)-F.{u) = S
J n vsssl V •
♦ J. B. S. Haldane, Proc. Cambridge Phil. Soe. (28), 1932, pp. 234-243.
342 THE DIFFEREJSrCE EQUATION OF FIRST ORDER
by Taylor’s Theorem, Thus we have
00
s
s = 1
oo
"VT
[U-5
This is an identity. Equating coefficients, we have
=«5)’
(6) S w(»)]— >(») = o.
This determines successively the functions and the solution
is complete in so far as the above operations are valid. In terms of
the function Fs{u) this recurrence relation can be written in the
symbolic form
{4> [u)I>+F[u)Y ~ [u) Y Fq (ii) + Fs{u), s> 2,
where the index of F{u) is to be written as a suffix after expansion
and D denotes the differentiation operator.
From the recurrence relation, we obtain
Aiu) = l, = A{u)=-Yf-lr,
/s («) = J [ - 19 - 59 <!>" <!>"'+ .
Thus we have
^ Cu (x) 2
^ J «(o) ^ 2 ^ ~ .
_ i [<l>'{v)f+<f>iv)4"(v) ^
12j„(o) cf>{v)
The arbitrary element in this solution is m(0).
We give some examples taken from Haldane’s paper :
Example 1. A “ =
W{U, h) = M-^-'’ + P(c+l)M-l-^Vi;2c(c+l)(2c+l)«<’-l
+ 'sV^^ (<5+ l)^(3c+ 1) ... .
343
11. 5] THE DIFFERENCE EQUATION OF FIRST ORDER
and therefore
x = A~ + g (c+ 1) log ~ (c+ 1) (2c + 1 ) ku^
where ^ is a function of u{0).
Example 2.
w (u, k) = + ^ka - jh kVe^^ 4- + . . . ,
and hence
p~au 11 1
X = A j kae^'^-\-,x .
ak 2 12 o
11 *51. Boole’s Iterative Method. When the in(le|)(‘ndeni
variable is absent, Boole writes the equation in the form
whence
^35+2 ”
where ^^{Ug.) denotes ^ [^'(^a:)]-
Proceeding in this way
'i^x+n = ^^K)*
If we assume an initial value to be known, we have thtu'efore
^a+n = +^‘K)-
It is evident, apart from the difficulty of application, tliat this
method is only suitable for a variable which differs from tlu) initial
value a by a positive integer.
Example 1. = 2uJ^,
We have = 2 (2^^.^)^ - 2*^ w/,
and, continuing, we obtain
344
THE DIFFEBENCE EQUATION OF FIRST ORDER [11 ‘51
SO ttat has these values in order according as w = 1, 2, 3
(mod 3).
11 -6. Solution by Differencing. Consider a non-linear
difference equation of the first order
(1) f{x, u, = 0.
Writing where t; is a function of x, and operating with
A, we obtain a relation of the form
X, V, A^) = 0.
If this be independent of w, and if we can solve this difference
equation for v, we obtain
(2) i;, cj) = 0,
where w is an arbitrary periodic, the elimination of v between (1)
and (2) will yield a primitive of (1).
This method may, in particular, succeed when (1) can be solved
for u in terms of x, v.
Consider, for example, the form, analogous to Clairaut’s differ-
ential equation,
Writing = v, we have
Ug. = xv+f{v).
Operating with Aj this gives
0 = (a?+l) A^+/(^ + A^)“/Wj
whence either A ^ = 0 or
(3) x+l+i^^±Mzm=.0.
A^
11.6] THE DIFFERENCE EQUATION OF FIRST ORDER
= 0 gives V = w, so that we have the primitive
Ug, =
The supposition (3) may lead to a second primitive.
Example, A + (A
Here
Ug, =
whence, operating with A>
(a?+l)A^ + 2^; A^+ (A^)^ = 0,
whence either ^ -y ;= 0 so that
or
which gives
Ug. = xm-^- txy^,
/S.V + 2V + X+1 = 0,
%+l-^'Og,^ -X-l,
the solution of which is easily seen to he
345
Eliminating v between this and the original equation, we have
This form of the solution may also be derived from the primitive
Ug, = cx+c^,
by supposing that c is a function of x and then taking the difference.
We thus obtain
^u= c+ (ic+1) A<^+2cAc+( A<5)^-
On the other hand, the supposition that c is a periodic gives
Equating these values of A hsive
Ac(ir-|-l + 2c+ Ac) = 0.
The equation Ao — 0 leads back to the original primitive, the
supposition that
x4-1+2c+A<^ =
gives the second form obtained above. Boole gives the name
indirect solution ” to a primitive obtained in this way.
a46 THE DIFEEBENCE EQUATION OF FIBST OBDEB [II.7
11*7. Equations Homogeneous in u. The general type
of such equations is
which on solution for leads to a linear equation.
Consider, for example,
= 0.
We have
whence
= 2u^ or
% = tiT 2® or = w.
11*8. Riccati ’s Form . The diiference equation corresponding
to Eiccati’s differential equation is
(1) u{x)u{x+l)+j[){x)u{x+l)-\-q{x) u{x) + r{x) = 0.
The substitution
u{x) =
'y(a:+l)
v(x)
-p{x)
gives the linear equation of the second order,
« (a; + 2) + [j (a;) - (a; + 1 )] «; (a; + 1)
+ [r(x)-p{x)q{x)]v{x) = 0,
the discussion of which does not belong to this chapter. We can,
however, obtain the solution of (1) when three particular solutions
are known. For let (x) be a particular solution of ( 1 ), and write
u{x) =
1
w{x) *
We then obtain, since t(^(x) satisfies (1),
This is a linear equation of the first order and therefore the
solution is of the form
w{x) = mf{x)-i-(l>(x),
where m is an arbitrary periodic.
Hence the complete solution of (1) is of the form
•
(2)
ll-S] THE HIFFEREJSrCE EQUATION OF FIRST ORDER 347
Now let ^2(^)5 ^3(^)5 ^4(^) be four particular solutions
of (1) and ^1, tU2j uTg, the corresponding periodics. Tlien it is
easy to verify, from (2), that
(^4 "• ^1) (^3 “ ^2) ^ (^4 - ^1) (^3 - ^2) ^ ^
K-^2)K-%) (^4“^2)(^3“^i) ’
say, where tu is a periodic. Thus the anharmonic ratio of four
particular solutions is a periodic. If we suppose to remain
arbitrary, equal to u say, we have
(U-U^){U2-U2) _ ^
{u-U2)(u^-u^)
which determines u in terms of the three known solutions and an
arbitrary periodic. The equation is thus solved.
11*9. Miscellaneous Forms. As examples of special artifices
which may occasionally be employed we cite the following :
Example 1 . '^x (^a+i - + 1 =
Here we have
This suggests substituting = tan v^, which leads to
Thus
and hence
tan A
A^a:
1
W'aj
tan"^ ^
= tan
m +
8
Example 2. ^
Here we put = cos v^, and we have
cos A
cos“*^ a^^x
and therefore
348 THE DIFFERENCE EQUATION OF FIRST ORDER [ex. xi
EXAMPLES XI
1. Find tlie difference equations to wMch the following complete
primitives belong :
(i) u = cx^+c^; (ii) m = {c( - ;
(iii) u = caj+c'a* ; (iv) u — cffl®+c® ;
(V)
&2a!+l
(T+^’
where in each case c, c' denote arbitrary periodics.
Solve the following equations :
2. = qa^.
3. u^^^—aUx = cos nx.
4. u^-y u^+(x+ 2) u^y + xux = -2-2x-x?.
5. Mjs+i - Wj, cos ax = cos a cos 2a . . . cos (as - 2)a,
X being a positive integer variable.
6. UgU^+y+au^+b = 0.
T. Wjg au^'\-l) — 0.
8. % = e®“.
9. Ma,4.isin!E9-Wj,sin(a;+l)e = cos (a; -1)6 -cos (3a; +1)6.
10. Mx+i-aWj, = (2a;+l)a*.
11. — 2 Mj,® + 1 = 0.
12. {x+lf{u^^y-au^) = a<‘{3?+2x).
13. = 4(m,)2{(mJ2+i)}.
14. u^^y =
15. u^lS.u^ = a;(AMa,)*+l.
16. (M*+i)®-3aVM*+i(M^)2+2a3a;»(«*)8 = 0.
EX. XI] THE DIFFERENCSE EQUATION OP FIRST ORDER 349
17. If Pk til® number of permutations of n letters taken K
together, repetition be allowed, but no three consecutive letters
being the same, shew that
APjc = (n®-n)-
iK.
a-
where a, |3 are the roots of the equation
a:2 = (n-l)(a:+l).
18. Solve the equation
Aw* = (w*+i)®-(w,)2,
by writing = u*.
19. Aw«+2mj, = -a;-l.
20. =
21. Apply Haldane’s method to the equations :
(i) Aw« = I:w*^
() “»+i-
in the latter case substituting
1 + ^.
Un
22. The equation
[Smith’s Pri^e.]
u =
2a; +1
^ 2a;+lj
has the compkte primitiye u = wx^+w^. Shew that another
complete primitive is
23. The equation
has the complete primitive w = cr a® + txy^. Deduce another complete
primitive.
360 THE DIEFEBESrCE EQUATION OP FIEST ORDER [ex. xi
24. If Un+x = m I (n+l), shew that
2. 4... I
■1).
, „ - mU or -TT — i i
1 . 3 .5 ... » 2 . 4 ... mO ’
according as n is odd or even.
26. From the difference equation
Un = n A«n +
obtain the indirect solutions
2.4... (w-1)
m
AWn’
1 .3 ...n
1.3...(n-2)’”^+2.4..:(m-l)C
, when n is odd,
1.3...(w-l) , 2.4...W _ ^
^ 2T4..;Tn-"2)c? + T7z:::{n-r)^^’
26. The equation
u = ic A^+(A
has the indirect solution
shew that, assuming this as complete primitive, the equation
u — t!3X’\'W^ results as an indirect solution.
27. Shew that the equation
iu = £iHAM)f_Aw
* 9 3
has the complete primitive
^ ^ _ ^3^ ^3^ [Poisson.]
28. Shew that the equation
^a:+l = (1+Wa.^)3
admits the complete primitives
(w+a;)», (’wa»--Ary,
\ a- 1/ '
where
a^ + a+l = 0.
29. Solve the equation
+ a-^) = 0,
by writing it in the form ( E - a-®) ( E - a^) = 0.
CHAPTEE XII
GENEEAL PEOPEETIES OF THE LINEAR DIFFERENCE
EQUATION
In tliis chapter we discuss properties which are common to all
linear difference equations and obtain some important general
theorems.
Many of the general properties are sufGiciently illustrated by con-
sidering an equation of the second or third order. Whenever this
method is suitable we shall adopt it.
12-0. The Homogeneous Linear Difference Equation.
The equation
(1) p„(a;)M(a;-(-M)-|-p„_i(a;)M(«+w-l) + ...
+J>i{x)u{x+l)+P(,{x)u{x) = 0,
where p„(a:)i given analytic functions of x
and where u {x) is the unknown function, is called a homogeneous
linear equation of order n. When there is no fear of ambiguity we
shall denote the coefficients by p„,
Ps{^)=Ps, s = 0, 1, 2, ...,w.
We have taken x to proceed by unit increment. The case of incre-
ment CO is readily reduced to this by the change of variable x = ya.
The homogeneous equation (1) has the triyial solution u{x) = 0.
We shall tacitly assume that this trivial solution is excluded from
all enunciations.
With regard to the coefficients ps(a;), we can assume that their
only singularities are essential singularities, for any poles can readily
be removed by multiplying the equation by a suitable integral
function which has zeros of the necessary order at these poles.
351
352 GENERAL PROPERTIES OP THE [12«0
We shall call the singular points of the difference equation the
three following sets of points :
(i) The zeros of denoted by aj, ag, ... .
(ii) The essential singularities of the coefiS-cients, P2, ... .
(iii) The zeros of Yi, Y2j —
The points
v = 0,1,2,...
will be said to be congruent to the singular points of the equation.
More generally, if a be any point, the points a + v, where v is zero
or an integer, positive or negative, will be said to be con^uent to
the point a.
If X be any point, and x-a he neither zero nor a positive or
negative integer, x is said to be incongruent to a,
12-01. The Existence of Solutions. Consider the second
order equation
Let us suppose that the value of u (a?) is assigned at every point
of the strip 0 < J? [x) < 2,
We have
+ Pi{x)u{x+l)
?o(a>) i>o(®) •
Hence, if a; be incongruent to the points a^-, we can find the
value of u{x) in the strip - 1 < < 0, and hence in the strip
- 2 U (ic) < - 1, and so on. Thus we can continue u{x) indefin-
itely to the left.
Similarly, we have
and if a: be incongraent to the points y,, -we can find u{x) in
the strip 2^2?{x)<3, and hence in the strip 3^fi(x)<4,
and so on. Thus, if we are given u{x) at every point of the
strip 0^i2(x) <2, we can continue u{x) over the whole of the
12-OlJ LINEAR DIFFERENCE EQUATION 353
remaining part of the complex plane except at the points which
are congruent to the singular points of the given equation. Hence,
if Wi{x)^ ^2{^) denote periodic functions of period unity such that
u (x) = m^{x), 0 < JS (x) < 1,
u {x) = m2(x), 1 < jB (x) < 2,
the above calculations will yield for u(x) a linear form in tai{x)j
m^ix) say,
U {x) — Wi (x) {x) -f tDTg {x) ^2 (x) .
The functions Ui(x) and U2(x) are particular solutions of the
difference equation which satisfy the conditions,
%(x) = 1, U2(x) = 0 in 0 < iJ (a;) < 1,
Ujl(x) = 0, U2(x) = 1 in 1 ^B(x) <2,
These particular solutions are in general discontinuous. Our
object is, of course, to find analytic solutions, but the above
investigation shews
(1) that the given equation has particular solutions ;
(2) that anal3diio solutions which satisfy arbitrary initial con-
ditions do not in general exist.
12*1. Fundamental System of Solutions. Let
u^{x), U2{x),
be n particular solutions of the general equation 12-0 (1). These
solutions are said to form a fundamental system (or set) when there
exists no linear relation of the form
where tUi, tjygr-*- 5 periodics such that for at least one value
of X, which is incongruent to the singular points of the equation,
they are finite and not all simultaneously zero. The functions of
a fundamental set are said to be linearly independent.
It follows that if {x) denote a member of a fundamental system,
and a a point incongruent to the singular points, that u^ia) cannot
vanish for all the points a, a+ 1, ... , a + n-1.
For if this were the case we should have Ug{a+m) = 0, where, m
is any integer positive or negative.
354 GENERAL PROPERTIES OF THE [12.1
If then denotes a periodic which does not vanish at the points
a+m, but vanishes everywhere else, we would have
w^Usix) = 0,
which is contrary to the hypothesis that Uq (a;) belongs to the system.
12*11. Casorati '$ Theorem . The necessary and sufficient
condition that % (a?), (a?), . . . , Un{x) should form a fundamental
system of solutions of the homogeneous equation of order n is that
U^{x) . . . Ur,{x)
u^{x+l) . . . Un{x+1)
I U2{x+n—l) . . . Un(x + n-l)
should not vanish for any value of x which is incongruent to the
singular ‘points of the equation.
For simplicity, take n^Z. The condition is sufficient. For if
u^[x), u^ix), u^{x) do not form a fundamental system of the equation
we can find a point ^ incongruent to the singular points of the
equation and periodics not all zero, when a? = such that
(?) + ®2 “2 (?) + WS (?) =
and consequently
+ 1) = ^3,
^iM^ + 2) + m2U2il+2} + w^u^(^ + 2) = 0,
and, since zzrg, do not all vanish, we have
i>(?) = 0.
The condition is also necessary. For, supposing that D (x) = 0,
let Ui{x), U^ix), U^{x) denote the cofactors of the elements of the
last row in D (x).
We shall suppose that these do not all vanish. Then, by the pro-
perty of cofactors,
u^{x) Uj^{x) + U2{x) V2{x) + u^{x) Vq{x) = 0,
Ui{x+l) Uj^(x)+U2{x+1) U2(x) + Uq{x+1) U^{x) = 0,
u^{xi-2) U^{x) + u2{x+2) V2{x) + u^{x^2) V^{x) = D{x) = 0.
the determinant
u^{x)
D{x) =
u^{x-\-Y)
12-11] LINEAR DIFFERENCE EQUATION 355
Now U^ix+l), U^{x+1) are tte cofactors of the
row, and hence
u^{x) U:^{x+1) + Uz{x) J/g (»+!) + Mg (a:) Z73(a;+1) =zB(x) = 0,
Mi(a:-f 1) Z7i(a:+l) + M2(a:-fl) C/'2(a;-f l) + M3(fl;-f 1) JJ^{x+\) = 0,
Mi(a; + 2) J7i(a;+l)-f M2(a: + 2) iTgCai-f l)-f M3(a; + 2) i73(a;-fl) = 0.
If we suppose V-^^{x) to he the cofactor which does not vanish,
the first set of three equations determines uniquely the ratios
TJ^{xy V^{xy
while the second set determines uniquely
172(a;+l) 173(2;+ 1)
17i(a;+l)’ V^{x+iy
and since in the two sets of equations the coefficients of the un-
knowns are the same, we have
^2(£±l)_t^)_
V,{x+l) _U,{x) __
E7i(a;+1)~ C7i(a;)“^3say,
where yg? Ts periodics. If we take Ts = we have,
therefore,
u-^ (cc) + m2 U2 (x) -f- zzTg W3 {x) = 0,
which shews that the solutions do not form a fundamental set.
If all the cofactors of the last row vanish we simply resume the
argument with n-l functions instead of n.
Example, It is easy to verify that the equation
u{x+2)-{<x-^^)u{x-^-1)^-ol^u{x) = 0, a=^ P
has the solutions a"®, These form a fundamental system, for
so that I){x) does not vanish for any finite value of x.
GENEEAL PBOPEBTIES OP THE
[12-H
356
Another system of solutions is
a* p® sin 2700,
for wMeh
Z) (a:) = a* p® (p - a) sin 2Tca:.
D (x) now vanishes whenever a: is an integer, so that these solutions
do not form a fundamental system.
The importance of a fundamental set of solutions lies in the fact
that every solution of a linear difference equation is expressible as a
linear function, whose coefficients are periodics, of the solutions of
a fundamental system. To see that this is so, consider the general
equation
p„u(x+n)+p„..iU{x+n-l) + ...+j)j^u(x+l)+pQu{x) = 0.
If iq(a;), u^ix), ... , u„{x) be a fundamental system of solutions,
we have
p„iq(a:+«)+p„_iMi(a;+%-l) + ...+po«i(a5) = 0,
p„M„(a;+m)+p„_iM„(a:+»-l) + ...+poW„(a:) = 0.
Eliminating the coefficients, it follows that
u{x) %(a;) u^{x) . . . u„{x)
M(a:+1) Ui(a;+1) ti2(a;+l) . . . ■M„(a:+1)
u(x+n) •!q{x+«.) «„(aj+«)
Since the solutions Ui{x), ... ,u„{x) form a fundamental set, the
minors of the elements of the first column are all different from
zero, provided that x be incongruent to a singular point, so that
periodics vs, ttq, ... , tu„ exist, such that
vsu{x) + vs.^u^{x)+vszUi{x) + .^. + vs„u,,{x) = 0,
with BT ^ 0. Obviously, every expression of the form
u{x)= tETj Wi(a:) + OT2M2(a:)+... + 07„«„(a:)
satisfies the given equation. Hence the result is established.
It follows from this that the problem of solution of a linear
difference equation consists in finding a set of fundamental solu-
tions, In the case of the homogeneous linear equation of the first
12-11] LINEAE DIPFEBENCE EQUATION 357
order, if ul{x) be a particular solution, the general solution is
v3u^{x) (cf. 11-1).
The above result can also be used to form a difference equation
with a given set of fundamental solutions. For example, the
equation which has the solutions x, x{x-l) is
u{x) X x(ir-l)
«(x+l) x+1 (x+l)®
u{x+2) x+2 (x-f2)(x+l)
= 0,
x(x+l) u{x+2)-2x{x+2) u{x+l) + {x+l){x*+2) u{x) = 0.
Here
D{x) = x(x-f 1),
which only vanishes when x = 0 or - 1, but these axe congruent to
singular points of the equation. The singular points are - 2, - 1,
1,2.
12*12. Heymann’s Theorem.* Casorati’s determinant 15 (x)
satisfies the bnear equation of the first order.
From 12-11,
Wl(x-t-l) «t2(x-fl) M„(x-fl)
Z)(x+1)= u„{x+2)
u^{x+n) u^ix+n) . M„(x-fm)
From the difference equation itself,
- — m(x) = — M(x+l)+^M(x+2)-f ...+u(x+«).
TPn !Pn JPn
Multiply the first w - 1 rows of the above determinant by
£l £2 £ndL
P» fn Pn
and add to the last.
^ W. Heymann, J. /. reine u. angm. Math. 100 (1892).
358 GENERAL PROPERTIES OP THE [12.12
TMs row then becomes
^2(3;), , -^U„{X),
Jr'Ti Fn Fn
SO thaty moving this into the first row, we have
(1) D(x+l) = i^^^^Dix).
jr «
It follows at once from this, that if ^ be not congruent to a
singular point, D (^) is simultaneously zero or not zero at all points
congruent to
/Again, solving this first order equation by the method of 11-1,
we have
(2) D(a:) = OT exp [ § log ^ ^ ^]-
The periodic tu will depend upon the particular fundamental set
which is chosen to form D{x),
An application of He3raiann’s theorem arises in the equation of
the second order when one member of a fundamental system is
known. By means of this theorem a second member of the system
can be found.
Consider the equation
j)^u{x-\-2)-{-Piu{x-{-1)+Pqu(x) = 0.
Let u^{x), u^ix) form a fundamental set and suppose to be
known. Then
D{x) = u^{x) u^ix+l) - U2{x)\{x+l)
= U,{x)U,(X+l)A^y
Thus
and B{y) is given hy (2), so that ^(x) is deteimined.
Thus, for example, the equation
x(x + 1)m(x+2)-2x(x+2)«(x+1) + (x+1)(3:+2)m(x) = 0
12-12]
LINEAR DIFFERENCE EQUATION
359
has the solution u{x) = x, and, from (2),
D{x) = roexp[ § log^Aa:]
= »ezp(log?^>)
by proper choice of c. Hence, taking gj = 1, we have
D{x) = a;(cc+l),
so that
X
U2{x) ^ ^ I j/\x^=: miX-{-x{X-^).
0
If we wish to find a particular second member of the set, we can
take = 0. Then
U2{x) = x{x-^).
12*14. Relations between two Fundamental Systems.
If
Uj^(x), u^(x), u^{x),
Vi{x), v^{x), v^(x)
be two fundamental systems, each solution of one system must be
expressible in terms of the members of the other system.
Thus, for example, we must have
V^ix) = t?y2.1%(^) + ^2,2^2(^) + --* + ^2,«^n(^)»
The periodics ^ are here not arbitrary, but depend solely on
the two fundamental systems chosen. Moreover, the determinant
^1, 1
^1, 2 •
• ^1, n
Cl =
^2, 1
^2, 2 •
• ^2, n
^n,2 • •
• n
360
GENEBAL PBOPERTIES OF THE
[12-U
cannot vanish. Conversely, if u^(x), form a fundamental
^stem and we take a determinant fl 0 of periodics, the system
%(rz;), , Vnix) also forms a fundamental set. The proof is simple
and is left to the reader.
12*16. A Criterion for Linear Independence.
Theorem. IJ n functions U2{x), ... , u^ix) be such that
lim = 0, s = 1,2,.
where r is a positive integer, then these functions are linearly inde-
pendent.
Suppose the foiictions all to exist in a half-plane limited of the
left, and suppose if possible that they are not linearly independent,
that is to say, that a relation of the form
VJ^U^(x)i-W2U2{x)-\-... + mnUn{x) = 0
exists where the periodics cji, ••• > Bot all simultaneously
zero, x not being a singular point. Suppose that the last product
which does not vanish is m^Uj^{x), so that
miU^{x) + ... + m^u„{x) = 0.
Write x+r for x and divide by u^{x+r)
Then
tZTi
‘<h{^+r) ^ ^ Ui(x+r)
(x+r) ^-"^u^ix+r)
-h ... + = 0.
If in this relation we let r oo , we have, from the enunciation,
^ u,{x+r) u,+^{x+r) u„_:^{x+r)_
T-»««.+i(a:+J-) Ms+2{»+»’)"‘ Um(x+r} ’
s = 1, 2, ... , m- 1. Thus we have
^m = 0,
which is contrary to the hypothesis. Thus a relation of the form
stated cannot exist and the theorem is proved.
We shall later make applications of this theorem to deduce the
linear independence of solutions of a diflference equation from their
asymptotic forms.
J2-2] LINEAR BIFFERENCE EQUATION 361
12*2. The Symbolic Highest Common Factor.* Consider
the linear expressions
(1) P{u{x)'] = u{x+2>)’Yf^[x)ii{x~{-2)
+ J)i(x)u{x+1)+J)q (x) U (23) ,
(2) Q\u{x)] — q^{x)u{x-\-2i) + q2{x)u{x+2)
■¥q^{x)u{x-Vl)^q^ (23) u (23) .
If we perform on Q [u (23)] the operation E of 2-4, we get
(3) E 6[^(^)] = ?3(23+l)^^(23+4) + g2(23+l) u(23+3)
+ g^l(23-l- 1) (23 + 2) 4- ^0(23+ 1) W (23+ 1 ) .
If we multiply (2) by ^^(23) and (3) by ro(23) and subtract from
(1), we shall arrive at an expression of the form
Qi (^) ] = t2{x)u{x + 2) + t^{x)u{x+l) + tQ (23) u (x) ,
in which u (23+3), (23 + 4) do not appear, provided that ^0(23), ri(23)
be so chosen that
(4) ?4(^)--^o(^) ^3(^+1) = 0,
Tz [^) ■“ ^0 (^) ?2 1) - ip) % (^) = 0.
Supposing this to have been done, we may write
P[^(23)]-{ro(^2^) E+^i(^2>-)}C[^(^2J)] =
or, symbolically,
Piu{x)] = R,{Q[u{x)]} + Q,[u{x)l
where { } is put for the operator ^0(23) E +^i(^)-
Evidently, for the more general expressions,
(5) P[u{x)]= f,{x)u{x + s),
8 = 0
m
(6) Q[u{x)]= ^ q^ix^uix+s), n'^m,
8 = 0
we can find an operator
* Pincherle and Amaldi, Le Operazioni Distributive, ( Bologna, 1901), chap. s.
362 GENEBAL PBOPEBTIES OP THE [12.2
where the functions ro(a;), r^{x), are determined by equations
of the same type as (4), such that
(7) P[u{x)] = Ri{Q[u{x)'\}JtQj^[u{x)'\,
where
Qi[u{x)]= 2 ts{oo)u{x+s).
5 = 0
The order of Qi[u{x)] may, of course, be less than m-1 since
the coefficients may vanish identically.
Now, if the difference equations
(8) P [u[x)] — 0, Q [M(a:)] == 0
have a common solution u^ix), it is evident from (7) that u^{x)
must also be a solution of
ei[M(a;)] = 0.
Thus every solution which is common to the equations (8) is
common also to the equations
(9) Q [m (a;) ] = 0, \u {x) ] = 0.
Treating the expressions Q[u{x)], Qi[u{x)'\ in the same way,
we can obtain
Q[u{x)] = R2{Qi[u{x)]} + Q^\u{x)],
so that any solution common to (8), and therefore to (9), is common to
Qi{u{x)'\ = Q, Q^[u(x)] = 0.
Proceeding in this way we continually lower the orders of the
difference expressions, so that after a finite number of steps we
must arrive at a pair of equations, say,
Qk [u {x) ] = 0, [u (x) ] = 0,
which have in common aU the solutions common to (8), and which
are such that the process cannot be continued. Thus the process
must terminate with either
W Qfc+i[M(a;)] = 0,
(®) Qk+i [w {x)] = t (x) u (x) .
In case (A), we say that [u (a;)] is the symbolic highest common
factor of the expressions (5) and (6), and we see from (10) that the
12*2] LIJSTEAR DIFFEKENCE EQTJATIOK 363
equations (8) liave common solutions whicli are the solutions of the
equation
Qic[u{x)] = 0,
obtained by equating to zero the symbolic highest common factor.
In case (B), we can say that the expressions (5) and (6) are mutually
prime. In this case the only solution common to (8) is the trivial
solution u (x) = 0.
Corollary. If it so happen that all the solutions of Q[u{x)] = 0
satisfy P [u {x) ] = 0, we must have the symbolic relation
P[u[x)] ~ R{Q[u{x)]}.
12*22. The Symbolic Lowest Common Multiple. As it
is of some importance to ascertain whether two given equations
have any common solutions we now introduce the notion of the
symbolic lowest common multiple.
Consider two difference expressions (see 12*2 (5), (6)), P[u(x)],
Q [u (x) ] of orders n, m respectively. The lowest common multiple
of these expressions is the expression V \u{x)'] of lowest order such
that the difference equation Y[u{x)’\ — 0 is satisfied by every
solution of each of the equations P[u{x)] = 0, Q[w(cc)] = 0.
First, suppose P[u{x)], Q[u{x)] to be mutually prime. Then
7['i4(cr)] = 0 must be satisfied by the n solutions of P[w(aj)] = 0
and the m solutions of Q [u {x) ] — 0, and since these equations
have no common solution, V[a(x)] — 0 must be of order m + n.
Then by the corollary of 12*2, we have
V[u{x)] = R{Q[u{x)]},
V[u{x)]^S{P[u{x)]},
where
R{ } = {ro(a;)E"+...+^n(^)},
S{ } = {So{x)E^+-- + sJx)}.
Hence, if we perform these operations and equate the coefficients
of u{x), u{x-\-l), ... , we obtain
364
GENERAL PROPERTIES OP THE
[12-22
(x) (a;) + Sm-l (^) Pi (» + 1) + 5„_2 (x) Po(x + 2)
= r„ (x) (x) + (a:) (a: + 1) -j- r„_^ (x) g-o (a: + 2) ,
Si(x)p^{x+m-l) + S(,{x)p^_y^(x+m)
= hi^)qm{x + n-l) + ro{x)q^_^{x+m-l),
So(®)P»(a:+»^) = 'ra(x)q^{x + n).
Thus we have m+n+1 homogeneous linear equations to deter-
mine the m + n + 2 unknowns r,.(a;), s^(a;). The ratios of these
functions can therefore be determined and we have the expression
for Y[u{x)\ save for a factor which is a function of a:.
We have supposed that P[M(a:)], Q[u{x)] are mutually prime.
If this be not the case, the equations P [u {x) ] = 0, Q [m (a:) ] = o
will have at least one solution in common, and V [w(a:)] will be
of order less than m+n. The same method may be used to
determine Y [u{x)], but now we must suppose that j-nfa;) sAx\
vanish identically. “ ^
Hence eliminating ri{x), s,.(a;) from the remaining equations, we
have the condition that P[M(ai)] = 0, Q[M(a:)] = 0 may have a
common solution, namely, the vanishing of the determinant
0 0 q^{x) 0 0 . . . 0
PiNPo(®+l)0 0 ... 0
0 0 0 ... p„(a;-fm-l) 0 0 0 ... q^{x+n-l)
This condition is expressed directly in terms of the coefficients of
the given equations.
Consider, for example, the equations
fMx)]
= u{x+2) - 2x{x+2)u{xJrl) + {x+l)(x+2) u{x) = 0,
g[u{x)]
- (^-l)u[x+2)-{Zx-2)u{x+l)+2xu{x) ^ 0.
12-22] LINEAR DIFFERENCE EQUATION 365
The condition for a common solution is the vanishing of
(a;+l)(a?4-2) 0 2a7 0
-2x(x+2) (cc+2){a;4-3) -3x + 2 2a;+2
x{x-{-l) -2(a5+l)(ir+3) x-1 -3a;-l
0 {a;+l)(a3+2) 0 x
Replace the second row by the sum of all four rows, and increase
the third row by twice the last row, then subtract twice the last
column from the second column and we then have a determinant
which clearly vanishes.
Thus the proposed equations have at least one solution in common.
We now proceed to find the Highest Common Factor. Multiplying
the first equation by (a;-l) and the second by x(x+l) and sub-
tracting, we get after suppressing a factor x^-x+2,
H[u{x)] ^ xu{x-\-l)-{x-{-l)u{x) = 0.
This must be the Highest Common Factor, since we know that
the equations have at least one solution in common. That this is
indeed the case is easily verified, for we can see at once that
f[u{x)]^{xE-{x+2)}H[u{x)l
The solution of H[u{x)] = 0 is u{x) = mx.
The equation f[u{x)] = 0 has already been solved by Hey-
mann's theorem in 12*12. We can use the same method to solve
g [u{x)] — 0. We have
u^ix) — -m^x-^-vyx
— 'm-^x-\'-mx
8 ‘”8
Q 1 2='r(i/)
u y{y+^) r(y-i)
a/
i-A\
Ly
— w^x-Vwx
Q
O W+l
Tliiis the primitive is
u{x) = oTia;+® . 2“.
366 GBNEBAL PROPERTIES OF THE [12-24
12*24. Reducible Equations. A homogeneous linear differ-
ence equation whose coefficients are rational functions of x is
said to be reducible when it has solutions in common with an
equation of lower order whose coefficients are likewise rational
functions of x. An equation with rational coefficients which lacks
this property is said to be irreducible.
Given two difference expressions P [w (a;) ] of order n and Q\u{x)']
of order m ( < 9^), we saw in 12*2 that we can form the operator
} = {’•o(a=) +
such that
If the coefficients of P and Q be rational functions, so also are the
coefficients of R and Q^.
Also, if P[i6(a;)] = 0 and Q[tt(a;)] = 0 have a solution in com-
mon, this solution satisfies Qi[u{x)] = 0. It follows that if the
equation Q [u(a;)] = 0 be irreducible, the expression Qi[u(x)] = 0
must vanish identically and we have
P[u{x)] = R{Q[u{x)]}.
Hence P['ii(cc)] = 0 is satisfied by every solution of Q [u(x)] = 0.
Thus we have proved the following :
Theorem. When a homogeneous linear equation with rational
coefficients has one solution in common with an irreducible equation
whose coeffiicients are likewise rational, then the given equation admits
every solution of the irreducible equation.
Let P[u(£c)] = 0 be a reducible equation. By h}rpothesis there
exists an equation Q \u{x) ] = 0 also with rational coefficients which
has solutions in common with P [u (x) ] = 0.
If P[v.(£c)] = 0 denote the symbolic highest common factor of
P and Q, the solutions common to P = 0, Q = 0 also satisfy P = 0.
If P = 0 be irreducible, aU the solutions of P = 0 belong to P = 0.
If P = 0 be itseK reducible, we can continue the process until we
arrive either at an equation of the first order or an irreducible
equation. We regard an equation of the first order as irreducible.
Thus we have the following :
12-24] LINEAR DIFEERENCE EQUATION 367
Theorem. Given a reducible equation, there exists an equation
of lower order all of whose solutions belong to the given equation, and
there exist one or more irreducible equations all of whose solutions
helofig to the given equation,
12'3. Reduction of Order when a Solution is known.
Let u-^{x) be a known particular solution of the general equation
(1) P[u{x)]=j)^u{x + n)+f^^T^u{x-\-n--l)-\-,..-\-p^u{x) = 0,
so that
^K(^)] = + + = 0.
Make the change of dependent variable
u{x) — u-^{x) v{x),
so that
Pn'^i{^+'f^)v{x-{‘n)+pn~ivi>i(x+n-l) v{x-{^n- 1) + ...
+PqU;^{x)v{x) = 0.
Now, by AbeFs Identity, 10-07 (I), taking
= 75sWi(aj + s), b, = i;(a;-f5),
we have
-qn-iA'^{^-^^-'^)-qn-2Av{x+n-2)~ ..,-qQ/\v{x)
■i-v{x + n)P[u^{x)] = 0,
where
qs = To + Pi^i {x+l)-h,..+j)sU^(x+s).
Now P [% (ii?) ] = 0 by hypothesis. Hence, if we put
= vu{x),
we have an equation of order 1 to determine w{x), namely,
(2) + + +
We note incidentally that
2o = 7^0 % qn~i = -pn'i^i{x+n).
If W2(x) be a particular solution of (2), we have
= A®2(®) = W2(^).
368
so that
GENEBAL PBOPERTIES OP THE
[12-3
u^{x)
w.
it) At,
which gives the corresponding solution of (1).
Suppose now that we know m linearly independent solutions of
(1), say, Ui{x), U2{x), u^{x). Then (2) has the m-1 solutions
A A ^m(^)
^ Ui{x) ’ ^ u^(x) ' ^ (x) ‘
These solutions are themselves linearly independent, fox if we
had a relation
we could deduce
5 = 2
Uj_{x)
= 0,
Ml (a:)
which would contradict the hypothesis that the m solutions are
independent.
If m>l, we can therefore proceed by the same method to
lower the order of (2) and we can in this way depress the order
of the original equation (1) by m units.
In particular, if we know one solution of the equation of the
second order, we can reduce the equation to the first order and
hence complete the solution by the methods of Chapter XL
Example. The equation
(2a;-l) u{x+2)-~{^x-2) 1) + (6a;+ 3) u{x) — 0
has the particular solution Ui{x) = 3^.
Putting u{x) = the equation becomes on application of
the foregoing method
or
IL'(X I 1^ - •
12-3]
LINEAB DIFFEBENCE EQUATION
369
whence
so that
3iB “ 2^-^ J
Alternatively, {x) = x can also be used, as tbe constant bas no
special significance.
12*4. Functional Derivates. If P be a distributive operator,
that is to say if
P[u + v] = P[u] + P[v],
Pincherle * defines tbe functional derivate P' by tbe relation
P' [u] = P[xu]--xP[u],
Tbe second derivate P" [u\ is defined by
P"[^] = P' [xul-^xP' M
= P [x^u\ -2x P [xu\ 4- P [24] ,
and generally
P(n) ^ p(n-l) ^ p(n-l) [^] ^
and it is easily proved by induction or by direct substitution that
+ (- 1)^ x'^P[u\,
If in this relation we put in turn for n tbe numbers
n, n- 1, n-2, ... , 2, 1, 0,
multiply tbe resulting equations by
' Pincherle and Amaldi, Xe Operazioni Distributive, p. 189.
[12-4
370 GENERAL PROPERTIES OP THE
and add the results, we get
P [x^ u] = [m] + Q X P<«-1) [m] + (2) 2^2 P("-2) [m] + . . .
+ P [ti] .
12*5. Multiple Solutions of a Difference Equation. Let
Plu{x)] = + U(x + n- 1) + . . . ^ (^)-
Consider the difference equation
P[u(cc)] = 0.
If a solution Uj^{x) exist, such that
Ui{x), xui{x), x^ii^(x), , x^"^Ui{x)
are all solutions, the solution %(ir) is said to be a solution of
multiplicity v. Since, from 124,
P[a;'-iM(a:)]=p(-i)[M(cc)]+(''~ ^)a:P(''-2)[M(:c)] + ... +x''-^[Pu{x)l
it follows that the necessary and sufficient conditions that u^{x)
should be a solution of multiplicity v are that (x) should satisfy
each of the equations
P[u(a;)] = 0, F[u{x)]=^0, ..., PC-D [^^(^r)] = 0
and that P^’'H%(^)] i=
Evidently, then, the condition that the equation P[u(x)] = 0
should have at least one multiple solution is that the equations
P[u{x)]=^0, F[u{x)] = 0
should have a common solution. If H \u{x)] denote the symbolic
highest common factor of the expressions
PM^)l P'[u{x)l
the equation H[u{x)] = 0 will have as solutions all and only the
multiple solutions of P [t^(cc)] = 0 ; and these multiple solutions will
appear in H\u(x)] = 0 with one less order of multiplicity than in
the equation P['w(a;)] = 0. Suppose that P['M(a;)] = 0 has r
multiple solutions of orders Vj, Vg, respectively, where
vj, vg, are arranged in non-descending order of magnitude.
If we can find the solution of multiplicity v^. For
12*5] LINEAR DIFFERENCE EQUATION 371
E [u (a?)] = 0 has solutions of multiplicities v,,_j - 1, and if we
find H2 [u{x)] the highest common factor of H[u{x)] and H' [u{x)],
i?2 [^(^)] = ^ will have solutions of multiplicities v^-2.
If Vy._2 = 2, ['^(^)] “ ^ only one solution, of multiplicity
v^-2. If ^r-i> we can continue the process, until we arrive
at an equation which has only a single multiple solution. We thus
reduce the problem to finding the solutions of an equation which
has only one multiple solution.
Suppose this equation to be Es[u{x)\ = 0, {s = with a
multiple solution of order - 5. This multiple solution will be a
simple solution of ['^(^)] = 0, which is of the first order
and can be solved.
The order of the original equation can now be depressed by
units, and if Vy._2 ^ v^_i, we can proceed to find the solution of
multiplicity v^_i and thus further depress the order.
Proceeding in this way we can find all the multiple solutions
up to the stage if any at which when the process just
described comes to an end.
To carry out this process it is necessary to form the derivates.
By the definition
P' \u{x)'\ = P[xu(x)'\-xP[u{x)'\
= {x-{-n)jpn'^{x^n)-\-{x + n- l)'p^_^u{x^-n-l)^
-f {x-\-V)'p-^u(X'\-l)-\-XfQU[x)
- X'p^u{x-{-n) - xpn-i^{^ + '^-P}-^Pi'^{^-\-^)-XfQu[x)
= n'PnU{x-{‘n)+ {n-l)'Pn-iU{x^n~\)-\- .,.-\-jp^u{x-^l).
+p^u{x+l),
and generally
H- 2’'^2 ^ +Pi
As an illustration, consider
P[w(a;)] = (2a;2 + 4a;+l) ^^(a3+3) - (2a;2 + 8a;+3) w(a;+2)
- {2x^-\-ix-2^)u{x-{-l)-\-(^x^^{-^>x^{‘^)u{x) = 0.
F [u{xy\ = ^{2x^ + ix+l)u{x■\-^)■-^%x^+Sx+^)u{^^
- {2a;2 + 4a; - 3) (a; + 1 ) .
372
GENERAL PROPERTIES OP THE
[12-5
We find
E-^P'lu(x)-]
^ ~ ^ ^ (^+ 1) - (^+ 1) ^(«^) }•
We can proceed to show that xu{x-{-l)-{x-{-l) u{x) is the highest
common factor. We can avoid the calculation by observing that
xu[x-\~l)-{x-\-l)u{x) has the solution u (x) = x, which is easily
seen to satisfy P\u{x)] = 0, P'[w(a?)] = 0. Hence P[^^(a;)] = 0
has the solutions x, x^. Knowing these solutions, we can depress
the order by two units and so obtain an equation of the first order.
The complete solution is u{x) — 1)*®.
12*6. Multipliers. If P[^^(a;)] be a linear form of order
a function M{x), such that M{x) P[u{x)] = /SiQ[u{x)]^ where
Q[u{x)] is a linear form of order n-1, is called a multiplier of
P[u{x)l
For simplicity, take
P[u{x)] = p2u{x + 3)+p2u{x+2)-\-piu{x+l)i'PQu{x),
and let •iti(cc), u^ix), Uq{x) be a fundamental set of solutions of the
equation P [w(a/)] = 0. ‘
Then, if u{x) be any other solution, we have
u{x) = miU^{x) + m2U2{x) + vj^u^{x),
U{X+1) = W-^U^{x-\-l)-^W2U2[x~\-l)-\-W^U^{x+l),
u{x-{‘2) = ^Z7l%(^r+2)4-tU2^62(cC + 2)^-tU3U3(^C + 2).
Solving these equations for tzTgs uxg, we have
^2(^)
^2(37+1) W3(a;+1) w(a;+l)
U2{x+2) t^3(x+2) u{x+2)
Uj^{x) u(x)
Ui{x+1) ^^3(^r+l) w{ir+l)
u^{x-\-2) u^{x + 2) u{X']-2)
1
Uy^(x)
^2(37+1)
tq(a?4-2) U2{x+2) u{x-^2)
Uj^{x) U2{x) Uq(x)
U^(x-j-l) U2ix+1) U^ix+l)
tq(x--|-2) U2{x-\-2) u^[x-]-2)
12-6] LINEAR, DIFFERENCE EQUATION 373
The last of these determinants is Casorati’s determinant D{x).
Denote by the cofactor of the element in the ^th row and Jth
column of this determinant, divided by D{x). Then
+ ^ Q^[u{x)],
^2 = \li^u[x)^r\Lfu{x^-l)^-\L^^^ = Q^\u[x)],
cTg = + +
Hence
A6i[w(^)] = 0, AQ2[^(^)] = 0, AQ3['^(^2?)] = 0,
provided that u(x) be a solution of P[u(x)] = 0, which we have
supposed to be the case throughout.
Thus taking Qi[u(x)], say, we see that /S.Qi[u{x)] = 0 has the
same fundamental solutioDs %(cr), ^2(^)3 P[i^(a?)]=0.
Hence the expressions A Qi [u(x) ] and P [u{x) ] can only differ by
a factor which is a function of x, and by comparing the coefiB-cients
of u{x-\-S), which are respectively and we have
F3
Thus + — ^>3 is a multiplier. Clearly we can prove the
same thing for
[x^*’(a;+l) jjj, [x^®*(a;+l) 4-J)3.
If we denote these multipliers by Vg, Vg, we have
AQi(aj) = ViP[w(aj)].
Writing this relation in full, we have
{x+l)u{x^- 1) 4- 1) w(a? + 2) + (xf^(ic+ 1) ia(cc4-3)
“ ^ (^) u(x)~]x P {x)u{x^- i)- (xf ^ {x)u{x + 2)
~ \ {])^u{x + 2>)-\-'P2.'^(x + 2)-\-'p-^u{x+1) + 'Pqu[x) },
so that, equating coefficients of u[x), u{x+l)^ ... , we obtain
'PQ{x)vy{x)=
j)i{x) \(x) - [il^^(a;+l)- iiP(a;),
\{^) = (Af\a:+l)-i^P(a5),
^33(2;) \(x) = [i,f^(a:+l).
374 GENEEAL PROPEETIES OF THE [12-6
In the first, replace cc byrr+S, in the second x by a: +2, and in
the third ic by cc+l and add. We then get the equation satisfied
by \{x), namely,
PQ(x-\-^)u{x-\-S)-i-pj{x+2)u{x-{-2)+p2{x-{-l)u(x+l)
+I)q{x)u{x) = 0,
and clearly Vg, Vg satisfy the same equation.
This last equation is called the adjoint equation of the given
equation P[u{x)] = 0. Introducing the operator Ej the given
equation can be written
[PsW E^+Pii^) E^+Ti{oo) E+Po{x)]u{x) = 0.
The adjoint equation is then
and we have the important theorem :
The multipliers of a given homogeneous linear difference equation
are the solutions of the corresponding adjoint equation.
We also see that the multipliers Vj^, Vg, Vg are the cofactors of the
last row of the determinant :
D{x-^1)
u^[x-\-T) W2(aj+2) u^{x + 2)
%(cc+3) ^^2(^+3) ^3(0; + 3)
each divided by D (x + 1 ). Thus we have
ViZ^i(a:4-l) + V2W2(aJ+l)+V3t^3(a;+l) = 0,
\u^{x + 2) + v^u^[x-^2)-{‘V^u^{x+2) = 0,
ViP3Wi(a;4-3) + V2^93t^2(^^+^) + V3;P3'ii3(a;+3) = 1.
12*7. The Complete Linear Equation. Denoting as usual
the homogeneous linear equation by P[^4((r)] = 0, the equation
P[u{x)']=f{x),
where f(x) is a given function of x, is called the complete linear
equation.
12-7] LINEAR DIFFERENCE EQUATION 375
Let Ui{x), u^ix), ... , Unix) be a fundamental system of solutions
of P [u (ic)] = 0, and let v (x) be a particular solution of the complete
equation, so that P [v(a?)] = f(x).
Then the general solution of the complete equation is
n(x) = x(x) + tUj^Uj^ix) -h m2U2(x) + + mn Unix).
The problem of the solution of the complete equation therefore
reduces to the problem of obtaining a fundamental set of solutions
of the corresponding homogeneous equation and a particular solution
of the complete equation. For simplicity, we again consider the
equation of the third order,
(1) ^3 '2/ (a:^+ 3) + P2 1) w (a;) =: fix),
and we suppose that we are in possession of a fundamental set of
solutions Uiix), ^2(^)5 u^ix) of the homogeneous equation
(2) pQuix-^3)-\-p2'^ix-\-^)+Piuix-\-l)+pQuix) = 0.
The multipliers v^, V2, V3 can then be found from Casorati’s
determinant of the given solutions, or by solving the adjoint
equation. Both methods have been explained in 12-6.
To find a particular solution we use Lagrange’s method of
variation of parameters. We seek to satisfy the complete equation
by putting
(3) V ix) = ix) ix) + ^2 ix) U2 ix) 4- ix) % ix).
As we have three disposable functions a^, Ug, we can make
them satisfy two additional conditions. We therefore assume that
'^(x+l) = aiix)uj^ix-i-l)-ha2ix)u2ix+l)-ha2ix)u^ix-rl),
vix + 2) = a^ix)ujix + 2) + a2ix)u2ix + 2) + aQix)u^ix + 2).
The conditions for this are
(4) Wi(a:;+1) A%{3^) + 'i^2(^+l) A%(^^) + %(ci?+l) = 0-
(6) %(aj+2) A%(^^) + «^2(^ + 2) A<^2(^) + %(^ + 2) A«^3(3?) = 0*
Again,
= aj^ix) U]^ix + 3) + a2ix) U2iX'i-3) + aQix) u^ixi-S)
+ Uiix+3)A^lix)+U2ix + 3) Af^2i^) + '^3i^ + ^)A^3ix).
376 GENERAL PROPERTIES OF THE [12-7
Substituting in tbe given equation for v{x-\-2), t;{£c + 3)
and noting that u^{x) are solutions of the homogeneous
equation, we obtain
(6) J53 % (a? + 3) A % (^) + i>3 ^2 (^ + 3) A Cl2 (^)
+7)oW3(rr+3) A«3(^)
Equations (4), (5), (6) are sufficient to determine
Aai(a2),
but as a matter of fact we already know solutions of these equations
from the property (given at the end of 12-6) of the multipliers
Vi, Vg, Vg, which shew at once that
t^a^ix) = V2 f{x),
A«3(^) = ^3
so that the required particular solution is
X X (C
v(x) = Uj_(x) ^ ^if(t)Ai+U2(x) ^ V2/(0 A^ + MsW ^
C C C
For example, consider
a;(x‘+l) 'ii{x+2)-2x{x^2)u{^x^l)^{x+l){x + 2)u{x)
= x{x+l){x-\-2).
A fundamental set of solutions of the homogeneous equation is
X, x{x-l), so that
(x+l)x
"'“■"'’“U+2 (*+2)(*+l) =<“' + l)(» + 2). ft = ^(«=+l)-
Tie multipliers are therefore
-a3(a: + l) 1 x + \ 1
D{x+l)
or
1 1
(a:+l)(a;+2)’ ar(a;+l)(a: + 2) ’
377
12-7] LINEAK DIFFERENCE EQUATION
Hence a particular solution is
v{x) = —X
X
t{t+l){t + 2)
{t + 1)
X
t{t+l){t + 2)
= -x g <At + ^(a:-l) § lAi
0 0
= - \xB<^{x)^-x{x~V)B-^{x)
~ iV + ^^)-
12'72. Polynomial Coefficients. When the coefficients of
the complete equation are polynomials, the search for a particular
solution can often be simplified by the following method.
Consider
(1) Pb^ipo)] + + ^ f{x),
where j? (x) is a polynomial of degree m, and the coefficients
are polynomials of degree {x at most where \i ^ m.
Put
u{x) = aQ + a^x+ ... + a^^^j,x^~f^ + w(x).
Substituting, we obtain
P[w{x)]^f{x) -fix),
where f{x) is a polynomial of degree m whose coefficients depend
upon the m- p + l constants We can in general
choose the constants so that the coefficients of x^, , xf^
on the right vanish, so that we are led to consider an equation of
the same form as (1), but with the right-hand member a polynomial
of degree [x ~ 1 at most.
In the case of the equation
fi n~l
qn(x) Au + q„-i{x) A u+... + qaix)u = q{x).
378 GENERAL PROPERTIES OP THE [12-72
where qs{x) is a polynomial of degree not exceeding 5 and q{x) is a
polynomial of degree m, we can in general find a particular solution
by assuming that
uix) = ^o + ^i(i) + ^2(2) + ***+^w(^)
and equating coefficients.
12’8. Solution by Means of Continued Fractions.* The
homogeneous linear difference equation of the second order may be
exhibited in the form
(1) ^
being given functions of the variable x whose domain is the
positive integers including zero. It is assumed that a^, do not
become infinite for any value of x in this domain. The general
solution of (1) is a function of x, containing two independent
arbitrary constants, which when substituted for in (1) renders it
an identity. The general solution is a homogeneous linear function
of the arbitrary constants which we shall take to be the initial
values of in this case,
Denote by ^ the ccth convergent of the continued fraction
a I ^2
^ ^2 + a3 +
Then
= C^xJPx—l'^^xPx~2y
9.x = ?a;-3 + ^x 9x-2 •
It follows that Pa and are particular solutions of (1).
Now
P2 = a2Pi + hPo-
If we regard p^ and Pq as arbitrary and denote them by % and Uq,
we have
P2 = +
Assigning an arbitrary value Uq to p^ is actually equivalent to
writing for and 62 '^0 ^2*
* L. M. Milne-Thomson, Proc. Royal Soc. Edinburgh^ li (1931), 91-96.
12-8]
LINEAR DIFFERENCE EQUATION
379
If we write for the (a;— l)t]i convergent of the continued
fraction — ^ , we have
a ~ 0’
and it is seen that is derived from by writing for and
(3a._i for 62, so that
px ~ '^1 + Paj-I *^0 >
and hence the general solution of (1) is
'Ugj = 0Ca;-l^l+ Pa:~l '^0*
We have thus expressed the general solution of (1) in terms of the
components of the (a;“l)th convergent of the continued fraction
— — - , which contains no arbitrary elements and which is
^2+ <23+
written down from the given equation. It will be observed that the
values of ba- for a; = 0, 1 are irrelevant.
It is proposed to generalise the above result to the homogeneous
equation of order m.
Milne-Thomson’s matrix notation, described for the two dimen-
sional fraction in 5*3, allows us to write the above result in the form
Such a matrix ^product is in fact equal to a matrix of one row
and one column, that is, a scalar.
This result is easily generalised, for consider the difference
equation
(2) = Clx^x—x~^^x^x—2'^ •••
and the square matrix (containing m rows and columns) which
is equal to
Ujj; 1 0 0 . . . 0
0 1 0 . . . 0
Ca; 0 0 1 ... 0
0
0 0 0 ... 1
io: 0 0 0. ... 0
380
GENERAL PROPERTIES OF THE
We call tlie matrix product
a generalised continued fraction of m dimensions.
Now consider the product
r Ti.tc fc
n/r T T T I 5^2, fc 3^3 J &
— J\J 2 ^ k
'1* Tc '2> k: ^3, k
Writing A + 1 for h, we have
^k+l ~ ^k Jjc+1 i
which gives, on forming the product,
Pi* k+l ~ ^k+lPl*k'^W+lP2, ••• + jk+lPm*k’
P2,k+l^Pl*k> Pz,k+l^ P2,ky ••• J Pm, k+l ^ Pm~l, k'
If then we write for Pi,^., it follows at once that the top row
of the matrix can be written
Pk J Pk~l j • • • J Pk-m+l >
and similar results hold for every row. Thus
J1J2 ••• Jn ~
Pn Pn—1 Pn-2
in-1 in-2
{^) Pn Pn—l "t ^n Pn—2 “t • • • “t jn Pn—mj
in ^n in—l'^^n in— 2"^ ... + Jn in— mi
Wn = ^n^n-l + ^n'^n-2 + ••• +^*71
We call p^, qni i '^n “the components of the nth convergent of
the continued fraction, and we have therefore for the crth convergent
— ^1^2 ••• ^x-1
L Jx I
12-8] LINEAB DIPFEKENCE EQUATION 381
It follows from (S) tliat parj ?a;j ••• > ^a; particular solutions
of (2).
Now, from (3),
Tm == ^ j
If we regard , Pm~i as arbitrary and denote them by
Wq , j ... , ^771*— i> 'V/ e bave
Pm = + +i7n^0’
which call be written
EXAMPLES XII
1. Form the difference equations whose fundamental systems are
(i) a®, a; a®, a^^a®;
(ii) a*, (g)®*
and explain why the result is the same in each case.
2. Complete the proof of the statement in 12-14.
3. Prove that the equations
P2-2/(a:4-2)+piU{a;4-l)+^o^(^) =
q;xU(x-\-l)-qQu{x) = 0
[e3:. xir
382 GENERAL PROPERTIES OF THE
have a solution in common if
(a:) go (2:) 2o 1 ) + i’l (®) go {^) gi (a; + 1 )
+?>oWgi(^>=)gi(»=+i) = 0.
4. Find the condition that
P2u{xi-2)-rpiU{x-^l)+PQu{x) = 0,
q2u{x + 2)-^ qiu{x+l)+ qQu{x) = 0
should have a solution in common.
5. Find the solution common to the equations
{2x^ +4x+l)'?^(a;+3)~ {2x^ + 8a; + 3) (a:; 4- 2)
-{2x^^ix-2i)u{x+l) + {2x^ + d^x-\-l)u{x) = 0,
u {x-}-2) - u {x-\-l) - 2u (x) = 0.
6. Given the expressions
P [u^] = {2x^ + 4x + 1 ) %+3 - {2x^ + 8a; + 3)
- {2x^ + 4a; - 3) + (2a;2 + 8a; + 7) ,
Q i'^x] = x{x+l) - 2a; (a; + 2) + {x+l){x-^2)u^,
prove that
PW = 22{QK]},
and shew that
T?f \ — 2a;2 + 4a;4-l p 2a;2 + 8a;+7
^ a:;3 + 3a;+2 ^ a;^ + 3a; + 2
7. Prove that the adjoint of the adjoint equation reproduces the
original equation.
8. Prove that the sum or difference of the ad joints of two linear
difference expression is the adjoint of their sum or difference.
9. If p {x), q [x) he rational functions, shew that the equation
u (a;+ 2) +p (x) (x+ 1) - g (x) { y (x+ 1) -y (x) } w (x) = 0
is reducible. Prove also that the most general equation of the second
order which is reducible must have the above form.
10. Obtain a fundamental system of solutions of the equation of
example 9.
EX. xn] LINEAR DIFFERENCE EQUATION 383
11. Prove that every equation which has multiple solutions is
reducible.
12. Prove that the equation
w(a:+2)-2^^M(a;+l) + ^i^M{a;) = 0
•4/ “r J.
is reducible.
13. Find a particular solution of the equation
3 2
A^+a;(a;-l) +
14. Given that a particular solution of
^a:+2 ltx+1 + ^a; = 0
is
deduce the general solution.
CHAPTER XIII
THE LINEAR DIFFERENCE EQUATION WITH
CONSTANT COEFFICIENTS
13*0. Homogeneous Equations. Consider tlie equation
P[u{x)] = j)„«(a:+w)+y„_iw{x+w- 1)+...
+'p^u{x+l)+pf^u{x) = 0,
where fn-i> To constants and =l=0,p(,^ 0.
There is evidently no loss of generality if we take = 1.
The equation can then be written
(1) P[w(a!)] = [E”+i>»-iE""^+-+J>iE+:Po]w(®) = 0-
Putting u{x) = g^v{x), we have
P[p®^)(a;)] = p*[p"E"+Pn-iP”"^E”~^+-"+FiPE+Po]'*^{®)-
Denote by /(p) = p"+p”~^Pn-i+-" + pPi+Po characteristic
function of the given equation.
Then
P[p*^(a;}] = p*/(pE)^>(«)
= p®/(p + pA)^’(4
since E=l+A-
Expanding by Taylor’s theorem, the equation is equivalent to
(2) [/(p)+p/'(p)A+|;r(p)A+-+5/(">(p) A]K^) = 0-
This equation is evidently satisfied by «;(a:) = 1, provided that p
be a root of the characteristic equation
(2) /(p) = P"+P”“^Pn-i+- + PPi+i?o = 0-
384
WITH CONSTANT COEFB'ICIENTS
13-0] WITH CONSTANT COEFFICIENTS 385
Let pi, P25..., Pn be tbe roots of the characteristic equation,
which we suppose to be all different.
Then we have n particular solutions of (1), namely,
Pl>
pf.-. p^
These solutions
form
a fundamental system,
determinant
p!
Pi
• • • p^
D{x) =
pf+i
pr'
• • •
p*+«-
■1 p|+«-i
••• P|+"''
1
1
1
pi
Pa ■ • •
Pn
= (piPa ••
• Pn)“
Pi
Pa
P»
pr
1 p-i ...
pr^
= [(-l)"l»o3*n (pi-p;), (see 1-5).
Since ^ - Pi ^ P? (^* =h3)y vanislies for any
finite x.
Example 1. u{x-{-2)-7u{x+l)-\-12u{x) = 0.
Tlie characteristic equation is
p2-7p + 12 = 0.
Hence
u {x) = 4®.
Suppose now that the characteristic equation has multiple roots.
Let pi, say, be a root of multiplicity v.
Then
/(Pi) = o, /'(Pi) = 0,..., /('-^>(Pi) = 0, /W{pi)=^0.
Putting p = Pi in (2), we obtain
386 THE LIHEAK DIFFERENCE EQUATION [13-0
We can satisfy this equation by taking v (cc) to be any of
I/v' /y*2
y iX/j tl/ y * * * ^ U/ •
Hence, corresponding to p = Pi, we have the solutions
pf, a;pf,
so that a multiple root of multiplicity v of the characteristic
equation gives a set of v particular solutions, and these solutions
contribute to the general solution the term
mj,pf-{-zff2Xpf+msX^pf+,,,i-m,_j^x^-^ pf = q^(x) pf,
where q^ix) is a ‘‘ polynomial in x whose coeiSSLcients are periodics.
Thus, if the characteristic equation have Ik distinct roots, we have
the general solution
u(x) = q^(x) p^+q^(x)pl+..,-h qj^{x) p|,
where the coefficients of the “ polynomials ’’ q{x) are periodics. To
shew that this is indeed the general solution, we must shew that
it is impossible to choose the arbitrary periodics in such a way
that, when they are not all zero, u{x) vanishes identically.
For simplicity, take the case of three distinct roots pi, p2, P35 and
suppose, if possible, that we can choose the periodics (not all zero)
so that
q^{x) pf+?2(a;) Pi+?3(a^) Pj = 0.
Writing x+1, x+2 for a:, we have
?l(®+l) Pi'‘'^ + ?2(a!+l) P2''‘^ + ?3(*) Pf^^ = Oj
52 (a: +2) f+^ + q^{x+2) pl+^ + q^{x+2) p|+2 = 0.
Eliminating pi“ pj®, ps®,
iM ?2(») iz{x)
Pi5i(a:+1) Mi{x+\) P353(a: + 1) | s 0,
P??i(®+2) p|?2(«+2) p|?3(2;+2)
that is
Pi?iW + Pi A5i(a:)
P? {^) + 2pf A ?i (®) + Pf A ?i {x)
13-0] WITH COHSTAKT COEFTICIENTS 387
Tlie coefficient of tlie highest power of x in this is
1 1 1
^ Pi P2 Ps
Pi P| P|
where mis a periodic which is not identically zero. The determinant
never vanishes, so that the coefficient of the highest power of x
cannot vanish identically and the supposition is untenable.
Example 2. u{x-\-i) - 2tu{X’\-l) - 2u{x) = 0.
The characteristic equation is
p®-3p-2= (p + lf(p-2) = 0,
SO that
u[x) = (m-^-\-m2^x){-lY+w^2^.
Example 3. u{x+6) + 2u(x~\-^6)-\-u{x) = 0.
The characteristic equation is
p® + 2p®+l = (p + l)^(p + eH)2(p + e~i’"^*)2 = 0,
so that each root is repeated once and
u{x) = {mi + xm2){-l)^+{m^+xm^) -^(m^ + xmo) e”
or in a real form
u{x) = (tETjL +a;zzT2)(-l)®+(tiy3 + a;?xr4)cos -f ( tijg 4- x tUg ) sin .
13*02. BooIe*s Symbolic Method. The general equation of
13*0 can be written in the operational form
f{E)u{x) = 0,
where /(p) is the characteristic function.
Thus, factorising /(p), we can write the equation in the form
(1) (E - Pi)'‘(E - P2)^ (E - = 0,
where p^, pg, p* are the distinct roots of the characteristic
equation. The order in which the factors in (1) are written is
immaterial, since all the coefficients are constants.
K we choose u{x) to satisfy
(2) =
388 THE LINEAB DIPPEEENCE EQUATION [13-02
we have a solution of the given difference equation. Since any
factor may he put last in the form (1) we obtain altogether
h equations of the type (2).
Now, by the theorem of 243, we have
(E - = P|{P» E - 9kY Pi “’“(a:)
= Pr*A[Pi"®w(a>)].
Hence, to satisfy (2), we must have
or
whence
= p|(tiTi+^?T2a;+... + t<T^_3
Treating each factor of (1) in the same way, we arrive at the
same solution as in 13*0.
Corresponding to a root p of multiplicity v of the characteristic
equation, we have the fundamental set of solutions
p% ajp^, ...,
By suitably combining these we obtain a second fundamental set
S'
1
(x-\\
\ 1
( 2
' (v-l)
so that we can write the general solution in the form
u{x} = +
which is sometimes convenient.
13*1. The Complete Equation. Let the given equation be
u{x-\-n)-\-pn-iu{x-{'n-l) + ..,+p^u{x+l)+pQu{x) = <l>{x).
As we have seen in 12*7, to obtain the general solution we need
only find a particular solution of the complete equation and add to
this the general solution of the homogeneous equation obtained by
13-1]
WITH CONSTANT COETTICIENTS
389
putting ^{x) zero. This latter may he called the complementary
solution.
Let/(p) he the characteristic function. If the roots pj, ... , p„
of /(p) = 0 be all different, we have the frmdamental set of solutions
pf; P|> ••• > Pn- obtain the required particular solution we use
Lagrange’s method of variation of parameters. As we have seen in
12-7, the solution is then
X X
C c
where Vj, Vg, are the multipliers corresponding to the funda-
mental set of solutions.
Forming Casorati’s determinant
p*+l
pi+' • •
' • 9T^
pf+2
i)(a;+l) =
pi+' • •
■ 9r-
p.+»
pl+» . .
■ ■ 9V^
we know from 12-6, since = 1, that Vg, are the
cofactors of the elements of the last row, each divided by D(a:-f 1).
Now we have, as in 13-0,
2)(2; + 1) = pf^ pl+i ... p»+i n (P,- pA (bi = 1, 2, ... , n).
0>i
Consider the cofactor of Clearly this is of the same form
as D{x+1), but formed from the elements p^, pg, ... , p„_i, and is
therefore
pr' - Pntl n (p- Pi), (ij - 1, 2, ... , n-1),
j >i
and thus
~ ^ ^ / n (p« “ P*)’ (^ == Ij 2, ~ 1),
i
= p„-*-V/'(p»)-
The same argument shews that
''/c= Pi*“^//'(ps)’ (*= 1, 2,..., «).
THE LUSTEAR DIFFERENCE EQUATION
[13-1
390
The required particular solution is therefore
ipfg
C
and thus the general solution is
«(x) = g
Pi-
Excmfhl. u{x+2)-u{x+l)-^u{x) = X.
Here
/(p) = p2-p-6 = (p-3)(p + 2),
so that a fundamental system of the homogeneous equation is
3®, (-2)®. Since /'(p) = 2p-l, the corresponding multipKers are
2,-x-x j _ 2)-»-i / 5, and the complete solution is
u{x) = Z^(r.,+ § ^:!^«Af) + (-2)®(«r,- g At) .
Now, from 8-1,
n
Taking c = 0, as is permissible since only a particular solution
is required, we have
= 2)^ - (2ct; + 1 - 20 .
__i_(3a;-l + 9(--2ri),
which is equivalent to
U {x) = 3^ + tU2 ( — 2)® —
where the terms 3®“^, “|■(-2)®~^ have been absorbed into the
terms cj2(-2)^. In fact the constant c contributes nothing
to the generality of the solution, so that we can always omit
any constant terms in the summation which may arise from the
particular value attributed to c.
391
13-1] WITH CONSTANT COEFFICIENTS
Emmple 2. m (a: + 2) - 6m (a; + 1) + 6m (aj) = 5®.
/(p)-(p-2)(p-3), /'(p) = 2p-5,
^ X
u{x) = ®i2®+®,3®-2® g g|lA«+3® g
Now
X
^ CL^ A. i = + constant,
whence we obtain
M(a;) = t^,2®+t^,3®-|-(|)%(|-i
u{x) = ?zrj2® + t?T2^®+‘§ 5®.
In the above discussion we have supposed the roots of the
characteristic equation to be distinct. The method is still applicable
if the equation present multiple roots, but the solution does not
assume the very simple form which we have just found.
We illustrate the method by considering the equation of the third
order :
u{x+ 3) - {2a + b) u{x + 2) + (a^ + 2ab) u(x+l) - a% u{x) = <f>{x).
The characteristic equation is
(p-a)2(p-6)=.0.
A fundamental system is a®, xa^, 6®, and therefore
a^+i 6®*^^
Z) (a; + 1 ) = (a; + 2) 6®+^ = 6®+^ (6 - a)^,
a®+^ (a? + 3) a®"^^ 6®+^
and the multipliers are therefore
a“®”2[(6-a)ir-~2a+6] 6-®-i
(6-a)2
)~-a ’ {b-af^
392 THE LINEAR DIFFERENCE EQUATION [I3.I
SO that
X
u(x) = + § a-*-^[{b-a)t-2a+b]<l>{t)
e
X
+ aja"’ ^ /S,t
c
c
Example^. M(a:+3)-5M(a:+2) + 8M(a:+l)-4w(a;) = 352®.
Here
/(P) = P®-5p^+8p-4: = {p-2)2(p-l).
A fundamental system is therefore
1, 2®, 352®,
and the corresponding multipliers are
1, -2-®-2(!c+3), 2-®-2.
Thus
X X
u{x) - ^ 2*^ Ay + (^2“T ^ ^(^ + 3) A^)2®
+ (,%+! ^ tAt)x2-.
c
The first summation contributes terms of the form 2®, x 2“, which
already occur in the complementary solution and can be omitted.
Taking c = 0, we have
u(x) = - 1 {i B^{x) + % Bz{x)}]2^ + [w^ + -} B^{x)]x2^.
Omitting from the particular solution terms which occur in the
complementary solution, we have finally
*1^ ~ "i” 2* (^2 "t ^^3 "f* •
13*2. BooIe^s Operational Method. The methods hitherto
explained have been of a general character and of universal appli-
cation in so far as the sums exist. The labour of applying the
13-2] WITH CONSTANT COEFFICIENTS 393
general methods even in simple cases may be very considerable.
We now turn to operational methods which considerably shorten
the work of finding a particular solution. Boole’s method, which
we now proceed to explain, is of particularly simple application
in three cases, namely, those in which the right-hand side of the
complete equation is of one of the following forms :
(I) a polynomial in x ;
(II) a-;
(III) multiplied by a polynomial in x.
The third form, of course, includes I and II.
If/(p) be the characteristic function, the equation can be written
in the form
/(E) M(a;) = ^{x),
where ^ {x) is a given function of x.
For finding the solution of the homogeneous equation the general
method is as simple as Boole’s, since both in practice merely
involve finding the roots of the characteristic equation /(p) = 0.
We therefore need only consider methods of finding the particular
solution. To effect this Boole writes
and proceeds to interpret the meaning to be attached to the
operation on the right.
13*21. Case I, </)(x) = m zero or a positive integer.
Writing 1 + A for £» t^e symbolic solution is
Now suppose that the characteristic equation does not admit
the root unity.
Then
/(I + X) = Uo + + • • • +
where ^ 0.
394
THE LIHEAK DIFFERENCE EQUATION
[13-21
If we expand
we get
1
fUTx)
in ascending powers of X as far as
/(i+x)
= bQ + bjX-h +
g{X) X«^+i
7(1 + X)
where g (X) is a polynomial.
Thus
1 == /(I + X)(feo + ^i ••• +^w +5^(X) X”^+^.
Since the expression on the right is a polynomial in X we can
associate with it a definite operation, which is equivalent to unity,
by writing A
Thus
2 m w+1
a:>«= /(1 + A)(&o+^iA+62A+ — + A)a:“+Sr(A) A a:”
Now
Hence
m+l
A
0.
(6o4-&i A + f^2 A + ••• + A)
satisfies the equation
/(1 + A)^(^) = or =
and is therefore a particular solution of the equation. The actual
expansion of can as a rule be most rapidly performed by
ordinary long division. An alternative is to express
/(1 + X)
in
partial fractions.
If cj>{x) be a polynomial of degree m the same method obviously
applies.
Example 1.
u{x-\-2)'{-u{x-['l) + u{x) = x^ + x+1.
Here /(p) ~ p2+ p+ 1, and therefore
1 1
/(1 + X) X2 + 3X+3
, i.X+|-X^+ ... .
WITH CONSTANT COEFFICIENTS
395
13-21]
A particular solution is therefore
u{x) — (-3" — A + f A) (^^ + ^33+ 1)
If the characteristic equation admit the root unity of multiplicity
r, our equation becomes
fiiE)(E-iYu(x) =
where /^(p) is a polynomial of degree n-r in p.
Putting
(E - 1)^^ {^) =
the equation for v {x) is
fi{E)'o(x) = x^.
Now
/i (1 ■+■ X) = Co + Cl X + C2 X2 + . . . + c^_^ X^-% Co i= 0,
so that we can apply the method already discussed and obtain
2 m
v(x) = (6o + 6iA + ^’2A+--- + &mA)a:’”,
which is a polynomial of degree m in x. If we write this in the form
(see 2-12),
a particular solution of the given equation is obtained from
A‘^u{x) - +
Since A (g} = ^ j) > lequired particular solution is
Example 2.
M(x+4)-5w(a;+3) + 9M(a:+2)-7M(a:+l) + 2M(a;) = a:®+l.
Here
/(E)=-(E-if(E-2),
396 THE LINEAR DIFFERENCE EQUATION [13.21
and the equation can be written
(A-1) = x^+1.
3
Putting A^(^) = have
v(x) = ^ (^ + 1) = (-1-A-A-A)(x®+1)
= -(x^ + 3x^ + 9x+14).
Thus we have
AuM = -6g;-12g)-13(5-14''*
and a particular solution is
The terms 1, x^ x^ belong to the complementary solution, so
that we obtain for the general solution, after reduction,
u(x) = 2® + tzTg + txf^x^-4ix^- x^ + tw x^ - rio
13*22. Case 11, ^(x) = a^. Here we have
f(E)u(x) = a^,
and, symbolically,
u(x)
Now, from 242, </>(E)(^^= so that, if a be not a zero
of /(E)> we have the particular solution
since
/■(F) _?1 =fMa^- fjx
If, liowe'ver, os be a zero of order t of f{p), we ha've
/(p) = (p-ar/i(p),
-f^
where
13-22] WITH CONSTANT COEFFICIENTS
Put u{x) — a*® (a:). Then our equation becomes
397
fi{E){E-aya‘‘v{x) = o“.
Using the theorem of 2-43, this gives
E) (o E - ay v{x) = a*,
whence, since A E - 1,
r 1
O' IS.V{X) = ^ , -ji=T ■ 1
1
r-1 =
/i(«E) /i(o + aA)’ Siia)
by our former method in 13-21. Hence
a~^r\ a~’'r\ fx\
/\v{x)
so that
f^%a)-fW{a) \Qj’
, . fx\ a-’'r\
w = U-
//(’•) (a)’
and the required particular solution is
^Oj-r ^(r)
u{x) —
Examplel. u{x + 2)-{-a^u{x) = cosmx.
We have cosmx — R{e'^^^) where R denotes the real part.
Hence the particular solution is
. / pmix \
= £2 + ^2 = R [^2^g2r i)
_ (2^ COS m £c+ cos m(a;- 2)
” a^ + 2a^cos2m4-l
Example 2.
Here
u(x+^)-^u{x-\-2)^-\2u{x+l)-^u{x) = 2^.
/(p) = (p-2)3 /<«)(2) = 3!,
u{x) =
a;(3)2®-2
3!
: 2-3
398 THE LINEAR DIEFERENCE EQUATION [13-23
13*23. Case III, ^(x) = a*R(x), where R{x) is a poly-
nomial of degree m, say.
/(£)«(») = a’‘E(x).
Put M(a;) = a‘^v{x). Then by 2*4:3,
a“/(a E)^’(*) = a’‘B{x),
so that the equation becomes
f{aE)v(x) = R{x),
which can be treated as in Case I.
Example.
(E -2)®(E - 1) w(a;) = a:®2®.
Write u{x) = 2“=u(a!j, then
8(E-l)®(2E-l)^’(a;) = a:2,
1
3?
8{2A + 1)‘
= i(l-2A+4A)a:2
= -|-(a^-4a;+6)
4\2/ 8\iy'''4V0
Thus
and the complete solution is
/
u(x) = rai-1-2® J^^-\-xw^+x^w^ + -^
1 /'x\ 3 (x
Z(x
•“ A
13*24. The General Case. When the right-hand member
of our equation is not one of the forms already considered we can
proceed as follows. Tor simplicity of writing, we consider the
equation of the third order,
/(p) = pH;?2p^+i5iP+^o-
WITH CONSTANT COEFFICrENTS
399
13-24]
Suppose /(p) to have a repeated root p^ so that
/(p) = (p-Pi)Mp-p3)-
Then expressing 1 / /(p) in partial fractions, we have
^ /(p) P-Pi (P-Pi)^'^p“p3’
and we note for later use that
(2) ^ (p ~ Pa) (p ~ Pi) + -B (p - pg) 4- C (p - ~ 1.
The given equation has the symbolic solution
Using (1), we write this in the form
(3) u^{x) =
and we proceed to justify this process by shewing that we can
interpret the terms of (3) in such a way that the resulting function
does in fact satisfy the given equation.
With regard to the interpretation, we first postulate that the
relation
implies that
(5) (E-w)’-i/r(a;) = ^(33).
With this law of interpretation, we have
f{E)u,{x) = {E-h}HE-9z)M=c)
= [^ (E - Pa) (E - Pi) + B (E - Ps) +c (E -
=
since, by (2), the content of the square bracket is unity.
We have here made use of the commutative property expressed
by
(E - Px)^(E - Pa) = (E - Pa) (E - Pi)^«.
Thus we have shewn that (3) does in fact satisfy the given
equation if the operations be interpreted according to (4) and (5).
400
THE LINEAR DIFFERENCE EQUATION
ri3-24
It remains to carry out the operation (4), in other words, we must
find a (particular) solution of
{E-niYi^{x) == cl>(x).
Using the theorem of 243, this becomes
(m E - [w® i^)] = <t> (x) ,
which gives, since A = E - U .
r
A (^) ] = <f> (ir) ,
whence, from 8-12,
X
yfr {x) = ^ j A t.
C
Thus to find a 'particular solution, of the equation
/(E) M (a:) = ^{x),
we express 1 //(p) in partial fractions ;
^ , V '' r ,
the particular solution is then
where c is arbitrary and may he chosen to have any convenient value.
The case m = 0 is an apparent exception.
If, however, m = 0, we have
/(p) == P*/i(p)
and the equation becomes
fi{E)u{x+k) = (f>{x).
Writing u{x-\-h) = v (jr), we have an equation of the type already
discussed. This case is really excluded, since we postulated in 13*0
that Pq ^0. IsTo generality is gained by the contrary supposition.
WITH CONSTANT COEFFICIENTS
401
13-24]
Example.
u{x-{-i)-2u{x-\-^) + 2u{x^l)-u{x) = ~ .
Here
/(p) = (p--l)3(p + l),
1 .1 ,+
/(p) 2(p-l)3 4(p-l)2^8(p-l) 8(p + l)’
and a particular solution [x) is therefore given by
«.w=i ')iA<
Also, we notice that the complementary solution is
ti7 ( - 1 ) ® + tl72 ^ + ^3 J
so that terms of this type may be ignored. Hence
u-^{x) = \x‘^ {x) ~ %x ^ (ic) + 1 ^ (x)
^lx^{x)-^l^{x)
+i^(a;) + T\5'W.
Ml (a;) = ^ (a;) -x+ -|) + g {x) ;
for g{x) see 11-31.
13*25. Broggi’s Method for the Particular Solution.*
Consider the equation
(1) P[u{x)^ = it (cr + n)+Pn-i^^{3^+ ^ = ^(^)>
where po =/= 0. The characteristic function is
/(p) = p"+2)„_ip’-^+...+2Jo-
Let
1
gi?) = = ao+aiP + «2p®+— •
* TJ- Broggi, Atti d. r. Acc. d. Lmcei (6), xv (1932), p. 707.
402
THE LINEAR DIFFERENCE EQUATION
[13-25
Tien
(2) (x.qPq=:1, ao^i + aiPo = 0, oco^Pa + aiPi + ag^Po == 0, ,
<XoPn_l + OCj^Pn_2+ ... +(X„_i Po = 0,
^s+lPn-l~^^s+2Pn-2'^ '••’^^s+nPo = 0, (s = 0, 1, 2, ...).
If tien
limsup ^1 j
^->00
be less than tie modulus of tie smallest zero of /(p) , tie series
(3) F {x) = ao^(x) + ai^(ic+l)4-a2<5i(cc + 2)+ ...
converges and F{x) is a particular solution of (1).
In fact
P\F[x)] = aQ^Q^(a3)-j- !) + .•. = <j>{x)
from tie relations (2). Now let
W = a® ^J;(a?).
We iave, from 2*5 (1),
‘l' (=» + «) = + (a:) + (j) A 4' (a^) + • • • + Q A 'I' (k) .
If we substitute tbis in (3), after collecting the coefficients of the
differences,
(5) F {X) = a- \g (a) ^ (x) + ^/(a)Ai> {^) + ~^ 9" (a) A 4- (^) + .-.] ,
where it is supposed that o is not a zero of /(p).
When 4j(a;) is a polynomial, the series for F{x) terminates and
F {x) is the product of a® and a polyn omial.
But the expression^ (5) for the particular solution can still be
used even when 4^ (x) is not a polynomial, provided that the series
converges, which will be the case in particular if
^ ffS'<‘M«)A'Kcc) <1.
It will be seen that the method is equivalent to expanding
l//{a+aA)
WITH CONSTANT COEFFICIENTS
403
13-25]
in ascending powers of A contains the justification of this
procedure when it is applicable. Broggi proceeds to examine forms
which lead to factorial series, but we will not pursue the matter
further.
13*26. Solution by Undetermined Coefficients. In cases
where the right-hand side of the equation has some particular form
it may be possible to guess the form of the particular solution and
obtain this by means of undetermined coef&cients. This method
will succeed in particular if <^(cc) = (a polynomial in a;). We
illustrate the idea with a few examples.
Example 1. u{x+2)-6u{x-j-l)i-4cu{x) — 10.
Try = c, a constant,
c-6c+4c = 10,
whence
0= -10,
u{x) = ^i(34-V5)®+c72(3-V5)»-10.
Example 2. ^fc(a;-i-2)-4^^(^r-l-l)^-4^^(a;) = cc2®.
Here 2 is a double zero of the characteristic function. Com-
parison with 13-23 shews that we should expect
u^{x) = 2®(^a-l-6aj+c(^2) + ^(3))'
The terms 2® (a 4- 6a;) will appear in the complementary solution,
so that
«i(x) = 2«(c(®) + czQ).
Hence we expect to find c and d, such that
1 jr (^'b2)(a;-f-l)a;^
404
THE LINEAR DIFFERENCE EQUATION [13.26
Since there is no constant term, c = 0, and we see that d =
satisfies all the conditions.
Thus
u{x) = {m^+xm^)2^+x{x-l)(x-2)2^-^.
Example 3.
3) + 2'i^(a; + 2) -j- 1) + i!^(2r) =
: +
x{x-+3) (ir~l)(a; + 2)*
The right hand = - - ^
rH-
X ic + 3 x~l x-i-2'
This suggests putting u^ix) ~
-4- -A- whence
X x-V
h 2a 2b 2a 2b a b
flj + 3 x-\'2 x+2 £c+l x-\-\~^ x'^ x'^ x—\
1_ Ji
X x + Z x-l xi-2'
(X — — Ij cj+26 = 1,
2a+b = -l, 6=1,
2(a+6) = 0.
These equations are consistent and are satisfied by 6 = 1, a =
/(p) = (p+i)(p2+p+i).
Hence , , i
U{X) = rai(-l)®+tiJ2(0*+OT2w2®+-pi_,
x{x-iy
-1.
where co.-co^ are the imaginary cube roots of unity. This example is
due to Markoff.
13»3. Particular Solution by Contour Integrals. We
consider the equation
(1) '^(^'^^)'^Pn-iu{x+n-’l) + ,^.-^PqU{x) = <^(x)j
whose characteristic function is
/(p) = P”+T«-lP”-^^-...+^Jo•
Let pi, p2, pj. be the distinct zeros of/(p). About each of
these pomts we describe closed curves, say circles, which are exterior
WITH CONSTANT COEFFICIENTS
405
13-3]
to one another. Denote by C the contour consisting of the aggregate
of the contours of these circles. We seek to satisfy (1) by
where g{p, x), regarded as a function of p, is holomorphic inside and
upon each of the circles round the points p^, pg, p^.
The above expression will satisfy (1) if we have
(4) Wi(a; + w) = A|^pa:+n-l£j^^p + ^(a;).
For, if this be the case we have, on substituting in the left-hand
member of (1),
4> (a;) + 2^1 p (P” P""^ + • • • + J’o) p*"^
which, from (2), is equal to
and the integral vanishes, since ^(p, x) is holomorphic by sup-
position. Thus we have proved that Vr^{x) satisfies (1), if (4) be
true and if
(6) =
Change x into x + 1, then
But, from (5),
'Wi(a;+5 + l) = s = 0, 1, ti-2,
THE LINEAR DIFFERENCE EQUATION
and hence, by subtraction,
m kL
p.«-. ^ ^
We can fulfil both these conditions if we take for g{p, x) a
solution of the equation
5f(p, a;+l)-5F(p, x) = p-‘‘4>{x),
for
s = 0, 1, ..., m-2,
since /(p) is of degree n and the residue at p = oo of pV/(p) is zero.
On the other hand,
since the residue is now unity at p = oo . Thus we have the theorem,
due to Norland :
The linear difference equation with constant coefficients
/{E)w{a:) = <i>{x)
has the particular solution
provided that g(py x) be a solution of
A*S'(p, aj) = p““^(a:),
which is holomorphic inside the contour G, which consists of a set of
non-overlapping circles each of which encloses one, and ortly one, of
the distinct zeros of the characteristic function /(p).
Emmple. u{x+2)-5u{x-^l)-hGu{x) =
r(!r + 1)
/(p) = (p-2)(p-3)
WITH CONSTANT COEFFICIENTS
and tlie equation for g((), x) is
which is seen by direct addition to be satisfied by
which is holomorphic in the neighbourhood of p = 2, p = 3.
Thus
-“•W = a r(S^
Now, the residue of
p*+i(p-3) p*+Mp-2) 3»+i 2*+i
^i(aj) = I
ir 1 1 1 , I
3 L r (a: + 1 ) ■*■ 3 r (a; + 2 ) ■*■ 32 r (a: + 3 ) .
ir 1 . 1 . 1 1
2Lr(a:+l) 2r(a; + 2)'2‘^r(a: + 3) "■J
_ e^3“-iP(a;; J) e42*-iP{aj ; i)
r(a) r(a)
in terms of the Incomplete Gamma Function of 11*33.
13*32. Laplace’s Integral. If <j){x) can be expressed
by means of Laplace’s integral in the form
^(a;)=f p»-i;j;(p)dp,
JL
where the path of integration L passes through none of the zeros
of the characteristic function /(p), we have the particular solution
p-i|gdp,
/(E)«i(a=)=f
for in this case
408 THE LINEAB DIFEERENCE EQUATION [13-32
In particular, this method can be applied whenever ^ (a:;) can be
expanded in a factorial series, for then ^(x) can be represented
by Laplace’s integral.
Example. Consider the equation of 13-26, Ex. 3.
We have, if R(x) > 1,
3 3 1 11 1
x[x + ^)'^ {x-l){x + 2) x-l'^x x + 2 a;+3
p®+^) dp
= f p®+^) <ip
Jo
Also, /(p) = (p + l)(p^+p + l), so that we avoid the zeros of
/(p) if we take for L the segment (0, 1) of the real axis. Hence
Ui{x) —■ [ /
Jo (i
p^-^(p+i)(i-p^;
(pHhl)(pHp+l)
(pa5-2_ pa:-l)
1 1
X- 1 X '
in agreement with the result obtained by trial in 13-26. The present
method, which shews why the trial succeeded, would be applicable
even if the coeficients of the given fractions were not equal.
13*4. Equations reducible to Equations with Constant
Coefficients. Consider an equation of the type
u{x^n)-rA^']^{x)u{x + n- 1) + At2 4* W 4(^~ 1) u{x^-n-2) + ...
■^Ani^[x)i^{x-\)'i^{x-2)...i^{x-n + l)u{x') = <j>{x).
log 4 A ^ ~ y^{x-n)-\- constant.
and
y^{x-n+l) = x(^“^) + log4(^“^+l),
gX (a:-n+l) _
WITH CONSTANT COEFFICIENTS
409
13 4]
If, therefore, we put
u(x) = v{x)^
the equation reduces to
'o{x+n) + A-^v{x-\-n-l)-\~...+An'o{x) = ^(x)e~^^^\
and when A-^, ^2> ••• > constants, this is an equation with
constant coefficients.
In the same way the equation
. 4'(^+l) ••• ^{x+n-l)u{x+n)
+ Ai^{x) . ij;(a;+l) ... u{x + n-l) +
+ An-^^{x)u{x-hl)-\-AnU{x) = ^{x),
reduces to
v[x-\-n)-^ A-^v[x-\-n-\)-{- ... + AnV{x) — ^{x)e^^^\
if we put
u{x) = v{x)
where
X
^ log 4* (^) A ^ = X (^) + constant.
Example.
u{x-\-Z)-\-a^ u{x-\-2)-\-a?‘^ u{x+\)-{- u[x) =
= a. a^. a^~^, = a^.a^,
X X
^ log a‘-3 A i = log a ^ (J - 3) A <
= (I log a) {x - 3) (a; - 2) + constant.
Put
u{x) — v{ic).
Then
v[x-{'2>) + v{x-\r^)'\-av{x-\-l) + a^ v{x) =
a particular solution of which is seen, by Boole’s method, to be
- d-S + a-i + ai+a^’
SO that
^Ha"-6«+6)
^i(^) = a-S + o-i+ai + a3-
410 THE LINEAR DIFFERENCE EQUATION [13-5
13-6. Milne-Thomson’s Operational Method."^ We now
consider an operational method of solution founded upon the
operator P""^, which was introduced in 2*6. The method is
applicable to those problems in which the variable x proceeds at
constant (here taken as imit) intervals from an initial value which
can be taken as zero without loss of generality. We then write
when there is no risk of ambiguity. Otherwise we can use the
notation
(2) P(»)% = Mx-1 + Mx-2+-+“0-
Then
A P~^ = Wx
but
P"^A^a! = A^aj-i+A^a5-2+-** + A'^o> and thus
(3) P“^AWi« = ^a:“^0-
Thus A == P"^ A and only if, — 0, in which case
no arbitrary elements are introduced and the operators Aj P^^
are completely commutative. It follows that tsT^u^ = P'^'i^a;, if
the result of each operation vanish with x.
Let Z be a given function of x. Consider the function defined
by the three conditions :
(i) (A-a)’'Ma. = Z;
(ii) Mo = 0 ;
(iii) contains no arbitrary elements.
Operating with p-^*, we get from (i),
so that we can write
(4) ^,= {P-a)-Z.
Now, from 2*62, we have
Z = (A ~ (^Y % = (1 + A [(1 + ^ J
* Milne-Thomson, Zoc. cit, p. 38.
WITH CONSTANT COEFFICIENTS
411
13-5]
and therefore comparing with (4)
( p - a)-*- Z = (1 + a)» P-- [(1 + a)-^-r X] ,
and hence, from 2-6 (6), we have the fundamental theorem of the
operator P, namely,
(5) {P-a)-^X{x) = {l + a)»=P(;^_,„)f ;^“^)(l + a)-^-X{j),
•where X{x) •is any function of x.
13' 51. Operations on Unity. Prom the definition
and, from 2-71, by repeated applications,
(2)
P-*^! =
Since a; is a positive integer,
(3) p-»-il = 0.
Prom 13-5 (5), putting X = 1,
{P-a)-n = (! + «)=' p-Hl + a)-*-^
(4) =[(l+ar-l]a-i.
Again,
(1-a p-^)(l + a P~^+a^ p-®+...+a“p-®)l
= (1 - 1 =
and hence
1 = (l + ap-i+...+a*p-”)l
(5) = (l+a)«,
from (2).
Differentiating r - 1 times with respect to a, we have
1,
(P - a)’’ ^ ~ C- 1)
TMs result can also be proved without difficulty by induction.
Again, from (6),
P
p 1 = (1 + ai)^ = (1 + exp {ix tan**^ a).
412
THE LINEAR DIFFERENCE EQUATION
[13-51
Heuce
p2
P^ + a'
2 1 = (1 + COS (aj tan“^ a),
-p^-^ 1 = (1 -f sin (x tan-^ a).
The operation 1 where (f> and \p- are polynomials, and the
degree of is not greater than the degree of can he interpreted
by expressing p.^p^ in partial fractions. If
partial fraction, we have
v^(p) ^
(P-a)’
be a typical
(1 + a)®
from (6). This is the extension to finite differences of Heaviside’s
Partial Fraction Theorem for the differential * operator p.
13>52. Operations on a given Function X. The inter-
pretation of X is given by 13-5 (5). Let
vP
X is given by 13-5 (5). Let
9^(P1
W)
'{P-ay-
i(P) x-s-A_x
1/^(P) (P-«y
To each of the operations on the right 13-5 (6) may be applied,
inother method may be used if Z be of the form (1 + aYf{x), where
f{x) is a polynomial. We can expand /(a;) in factorials so that
Z = (1
(l-f-a)®
* See H. Jefireys, Operational Methods in Mathematical Physics (1931), for
the corresponding theory of the differential operator.
WITH CONSTANT COEFFICIEN-TS
413
13-52]
Hence
^(P) Y- vg .^i(P) P
f(P) ^ ^ V^{P)(P-ar
1
wtich is interpreted in terms of operations on unity. For example,
9(P-1)
P + 2
2®** =
9(P-1)
P + 2
2®-2 + 2a:.2®
9(P-1)^ 8P , 2P
p+2"V(p-i)3+(p-i)2;-
24P 2P 2P ,
"(P-l)' P-1 P + 2^
= 3a;.2®+2-2®+i+2(-l)®.
13' 53. Application to Linear Difference Equations with
Constant Coefficients. The general equation of order n in one
dependent variable is
n n~l r
(1) + A • • • +<^r A • • • + <^0
where the a^. are constant and X is a function of x only. Since
A “ [(^ “1“ A) “ 1]^ '^x *^a:+r “ ^aj+r— * “h ( *” 1)^ ,
(1) can also be exhibited in the form
'^x+n ^n-1 '^x+n~l 4- . • . + 'l^x-i-r + • • • + Sq
This is the form which generally arises in practice. It may be
converted into the form (1) by the formula ^ta.+r = (1 + A)^^a;-
Taking the form (1), we obtain the operational solution in terms
of the initial conditions
r
A^O = 0 (^~0, 1, 2, ... , n-1)
by continually operating with P“^ until we arrive at an equation in
which the operation A does not occur. Bach operation with
introduces initial values and depresses the order by unity. The
final equation in which A does not occur is solved for in terms
THE LINEAR DIFFERENCE EQUATION
414
[13*53
of P. The interpretation of the operations gives the value of in
terms of the initial values.
The method, which is quite general and which applies also to
simultaneous equations, is best illustrated by examples.
The equation of the first order.
/S^u^ -au^ + X,
u.
= Mo (1 + «)* + (1 + P“n (1 + X] .
The, equation of the second order.
2
Aw»-(a+&) Aw»+a6Ma. = X.
Denote by Vq the initial value of A % :
A % - '«0 - (“ + ^) ("“x - Mfl) + X,
M* - “o - P '^’o “ P (® + ^') («x - “o) + P = P X,
,, -- P^^O-P(<^+^)Mo+P^n . 1 Y
- (P-a)(P-6) +(p-a)(P-6)^
= . P* ^Q-g^O . P , 1 f 1 Jl_ \ y
a-b P-a a-b P-6'^a-6\P-a p~bJ^
If a = 6,
_ K-g^o) P . P%, ^
(P - af ■ + p - a+(p3^* ^
= (Vq — a Ufy) a? (1 +
+ (l + a)*p~2[(n.a)-^-2X].
* When no operand is given, unity is to be understood.
13-54] WITH CONSTANT COEFFICIENTS 415
13*54. Simultaneous Equations.
= X,
= 0,
v,^-Uo+p-^av^= P-^X,
i>x-i^o+ P~^bu^ = 0,
.. _P^Uo-Pav,+ pX „ _p^v,-pbu,-bX
P^-ab ’ P^-ab
which can be interpreted as before.
13*55. Applications of the Method. Probability.
A coin is spun n times. The probability of its sJmoing head at
the first spin is p' ; while at any subsequent spin the probability
that the coin shews the same face as at the previous spin is p. What
is the probability that the coin shews head at the rvth spin ? *
Let Un-x be the required probability. Then Wq = p',
“n-i = T + (1 - iJ) (1 - Wn-a) ,
A w„-2 = (2p - 2) w„_a + 1 - p,
Un-P' = p-M2p-2)M„+p-i(l-p),
P ^{fip-2)'^ P - (2p - 2)
= /(2p-l)" + ^[(2p-l)«-l],
= I + (2p - l)"-^(p' - i) .
If ==: I*, this is I for all values of n and 'p.
Geometry,
A, B, C are three spheres each outside the other two, and a
point P is taken inside A. The inverses P', P" of P are taken in
B and C, The inverses of P' are taken in C and A, and of P" in
A and B, and this process is continued. Shew that of the 2" points
which arise from P by n inversions, | ( - 1)” + 1 2” lie inside A.‘\
* W. Burnside, Theory of Probability (1928), chap. ii.
f This problem is tal* m from an examination paper set by Professor W.
Burnside at the Royal Naval College.
416 THE LINEAR DIFFERENCE EQUATION [13-55
Let Un, "On, w„ be the number of points which lie inside A, B, C
respectively and which have arisen at the nth stage.
Then Mo = !> ~ “*0 =
Un+l “ “t j ^n-il “ "t > '^n+l “ “t '^n i
SO that AWn+Mn-Vn-Wn = 0,
-U„ + t:iV„ + V„-Wn == 0,
-W„-V„-AM’n-Wn = 0,
(1+P-1)M„- P-^V„- P-1W„ = 1,
- p-1 M„ + (1 + P-^) - p-1 Mn = 0,
-P"^ W«- P"^ «» + (!+ P“^M’n) = 0.
Solving for
„ _ P(P^+2P)
''”“(p+2)HP-r)
_ P , 2P
“3(P-l) + 3(p + 2)
= §2« + |(-l)".
Dynamics.
Two equal ferfectly elastic spheres of masses M, m (M > m) lie on
a smooth horizontal surface with their line of centres perpendicular to
a smooth perfectly elastic wall. The sphere of mass M is projected
towards the wall so as to impinge directly with velocity V on the
sphere of mass m. Find the velocities of the spheres after the nth
impact between them.
Let Un, Vn l>e the required velocities of M, m respectively measured
positive when towards the wall.
Here ^^o = F, 'IJq = ^
M (m„+i - m„) + m +?>„) = 0,
'^fi+l + ~ ('^n+l ~'0n) — 0.
That is
M A«B+»w(A+2)n„ = 0,
(A + 2)m„- A^n = 0,
M M„+m(l4-2p-i)D„-i(f7 = 0,
(l+2p-i)«„-t,„-F = 0,
13-55]
SO that
Now
where
WITH CONSTANT COEFFICIENTS
_{P^ + 2P)mV+pmV
{lf + m)p2+4mP + 4m’
^ (P^+^P)MV-P^MV
(ilf + m) p2 + 4m P + 4m’
(M + m) P^-f 4m P + 4m = (M + m)(P-a)(P-(3),
-2m + 2i sIMm
417
Jf -f m '
■2m-2islMm
M+m ’
_ (M 4- m) a -1- 2m P F (M + m) p -h 2m P F
{M + m) (a ”• P) P - a {M + m) (a — P) P - p ’
2M
Vn =
Un
ri^_ pzi
-P)LP-oc P-^pJ^
(M + m) (a
|-F[(l + a)^ + (l + p)-]
If COS 0
so that
2^ V m
M -m
M + m’
1 + a = cos0 + ^sin 0,
1 + P = cos 0 - i sin 6,
Un=^ V cos ?^0,
Vn = y siaw9,
and the total energy is |MF^, as it should be. It will be noticed
that when n0 first exceeds J ti the more massive sphere is moving
away from the wall.
Energy.
A particle starts from a point Aq with energy E and passes
successively through the points A^, A2, A^, . At the points A2r+j
it absorbs a quantity q of energy, while at the points A 2,. it loses
half its energy. Find the energy at the point A^..
418
THE LINEAB DIFFERENCE EQUATION
[13-55
If Mj, be the required energy,
%r+l ~ i ^2r-l
“2r = i“2r-2 + i?-
Hence Ma,+2 = 4 % + 1 P - ( - 1)“],
P -f 3
or 2A®%+4Am»+«* = ?p:^.
Ua = E, A'«'o = q-
Operating successively with P“^,
2EP^+iiE+2q)P , P+3
2P2+4P + 1 +(2p2+4P + l){P + 2)^
2(a+2)5-2(a+2)9 P 2(p+ 2) ^?-2((3 + 2) ? p
2(a-p) P-a 2{a-p) p-p
-kp + 2 + t?>
-2+72 ^_-2-72
9. J P — o “
where a :
After reduction,
«, = i(5-?)[(^-)V2[l-(-l)='] + [l + (-l)*]}]
+ k[3-(-l)"].
Hence for large values of x the energy is alternately q and 2q,
nearly.
The linear oscillator with discontinuous time.
The Hamiltonian of a linear oscillator of mass m, momentum p
and displacement q is
the equations of motion being
BE _dq _ p
WITH CONSTANT COEFFICIENTS
419
If we suppose a minimum time interval to exist so that time
can only increase by integral multiples of this minimum interval
cy, we may tentatively generalise the above equations into
^ q=:S-
^ m
where — '
Changing the independent variable to x == 1 1 g, these become
^ m
Ap = -Gkq,
These give
p+p-'^akq-po = 0.
where
o- I P
^ p2 + ^ ^ P2+ — ’
^ m ^ m
p = i” (j)g COS ^^-qoJmk sin
i / , Po ■
? = ’-n?oCOS-+^sm->
r2 = 1+-
The coefficients of r*" are in general not periodic, since -g- is
not in general integral. But we recover the ordinary periodic
solution for continuous time when a-^0. If we calculate the
Hamiltonian, we get
ciA
i^+ikoVr^ = Bor’'.
[13*55
420 THE LINEAR DIFFERENCE EQUATION
wMch increases with t. It may be observed that the function
remains constant and might tentatively be called energy. This
suggests replacing j) and q by
p' = q' = qr''^I<^.
13*6. Simultaneous Equations. In 13-54, 13-55 we have
seen how simultaneous equations can be solved by means of the
operator P when the variable- is an integer. In general, to solve
such equations in, say, two dependent variables, we could proceed
to eliminate one of them and then solve the diiference equation
satisfied by the other. We illustrate the procedure by an example.
Consider
+ = 0,
We write these in the form
(E-l)^a:+2E«^x = 0,
“2t^a; + (E--l)^a; =
whence [ (E - 1)H 4E] ~ 2E a-
or (E + 1)^ Wa; = - 2a^+'^.
Thus M*= (ro+ojcTi)
From the first equation,
and therefore
=(ro+i%+ ©1 k) ( - 1)*+ •
13’*7. Sylvester’s Non-linear Equations. The solution of
two types of non-linear equations has been deduced by Sylvester
from the solution of linear equations with constant coefl&cients.
Consider
(1) M»+n + Pn-l "^x+n-l + • • • 4“ J?o
where the coeflS.cients are constant. The solution is
(2)
u.
13-7] WITH CONSTANT COEFFICIENTS 421
where the are the zeros of the characteristic function, so that
aia2...a„ =
If we write down (1) for x+l, x + 2, ... , x+n and eliminate
Pi, • • • > Pn~i j we obtain the non-hnear difference equation
'^£c+n
'^x+n—x • • •
(3)
"^x+n+X
'^jd+n • • •
'^x+X
= 0.
'^x+2n
^iB+2n-l * * •
'^x+n
Since (2) satisfies (1), it also satisfies (3), but since p^, , Pn-^v
on which the values of the depend, do not appear in (3),
we may regard the values of as arbitrary. Thus (2) furnishes
a solution of (3), the values of being now arbitrary constants.
The formal character of the solution given by (2) will not be altered
if we replace the by arbitrary periodics. Thus we have a solution
which contains 2n arbitrary periodics.
The other type of equation is obtained by writing down (1) for the
values x-\-\,x + 2, + l and then eliminating p^, > i^n-i •
This gives
'^x+n
'^a;+n-l *
%+n-l
'^x+n~2 *
^ar+n+l
'^x+n
• '^03+2
"^x+n-X '
■
'^x+2n~-X
^a;+2n~2 * '
■ • '^x+n
%+2n-2
'^x-h2n-3 •
• • %+n-X
Calling the last determinant K{x), we have
^(x+l) — a^ag ...
whence
K{x) = m . (ociag ...
Now, using (2), we have for K{x) the determinantal product
®iai
X
OC2 • •
«r^
ar" ■
. ai
1
x+X
Wi OCi
. ^
X
af"^
. ag
1
_ x+n-X
Wi OCi
x-t-n-l
W2 OC2 • •
_ «a;+n-l
0^"^ • ■
. a„
1
422
so ttat
THE LINEAR DIFEERENCE EQUATION
[13-7
K(x)=- CJi ^2 ®«(«l “2 ••• ««)“ n
i>j
Comparing tlie two expressions for K{x), we see that the differ-
ence equation
^x+n-2
'^a!+2n-2 %+2n-Z
=
^x+n-1
where tjy is a given periodic, has the solution (2) where
are periodics and a^, ag, ... , a,^ are constants, which
can be arbitrarily chosen subject to the two conditions
- ©1 ... tcr« n = m,
i>j
a^L a2 . . . a„ = m.
Thus 71 of the constants can be replaced by periodics and we
have a solution involving 2n- 2 arbitrary periodics.
Example. 'Z/jg+2 ^ j
where ct is a constant or a periodic.
This can be written
and therefore has the solution
^a:+l
where
ap= 1,
so that we have
= 77T/Y®4-
aa'
-flc+2
(a2-l)2
where ar is an arbitrary periodic and a an arbitrary constant.
If this be a solution when a is an arbitrary constant, it is likewise
a solution when a is an arbitrary periodic. The solution may
therefore be regarded as containing two arbitrary periodics.
13*7]
WITH CONSTANT COEFFICIENTS
423
The equation can be regarded as arising from
whence
^a;+2 "t JP “t '^x “
SO that, eliminating p,
'^x+i^ - '^x '^x+2 = '^x+1 ~ constant.
13*8. Partial Difference Equations with Constant Co-
efficients. Let be a function of the two independent variables
X, y. Taking the increments of x, y to be unity in each case we
have, as in 2-105,
= u{x+l, y)-u{x, y), Aj,M = U{x, y + l)~u{x, y).
It is also convenient to introduce operations Ea?? Ey defined by
= u{x-\-l,y), Ev^^ =
Then
I+Ajb^Exj 1 + Ai/^Ei/*
It is clear that the operators /^y are commutative, that is,
Ax Ay ^ ~ Ay Ax
If then F (X, p) be a bilinear form in X, p whose coefiScients are
independent of x and y, a difference equation of any of the forms
(1) -^(Axj Ay)^ = 0> ^(Ex» Ey)'^^ = 0,
■^'(Ax, Ey)^ = 0, ^’(Ex, Ay)^ = 0,
is a partial difference equation with constant coeflB-cients. A more
general type of such equations is
■f'(Ax, Ay)u=f{x, y),
where f{x, y) is a given function.
We can obtain formal symbolic solutions of equations of the
forms (1) by the following device. We first replace ot Ey by a.
There results an ordinary difference equation in which a figures
as a parameter. Having obtained the solution of this we replace
a by the operator which a represents and interpret the solution.
The method will be understood by considering some examples.
424 THE LIHBAB DIFFERENCE EQUATION [13-8
Example 1. u{x+l, y)-u{x, y+l) = 0.
Tkis is equivalent to
Ea;^-Ei/^ = 0.
Writing for we have
a solution of which is
u = a^<j>{y),
where <^{y) is an arbitrary function of y and is written after the
symbol a®.
Thus
u-=(ByYj>{y) = <t>{x+y),
which clearly satisfies the given equation.
If ^i{x), ujgly) denote arbitrary periodic functions of x, y of
period unity, it is evident that
^i{pc)w^{y)j>{x + y)
is also a solution, which is more general, in that xj3-^{x) w^{y) is not
necessarily a function oi x-\-y. We can replace this product by
w{x, y), an arbitrary function periodic in both variables.
Arbitrary periodic functions can always be introduced in this way
into the solution of an equation of the types (1), but, for simplicity,
we shall ignore them.
Example 2. u{x-}-l, yi-l)-u{x, y+l)-u(x, y) = 0.
This equation can be written
Ev = 0.
Replacing Ej/ by a, we have
^ Aa; ^ = 0,
a solution of which is
where ^ (y) is an arbitrary function, again written last. Thus
(2) ^ = (1 + E;')^<ji(y).
13-8] WITH CONSTANT COEFFICIENTS 425
Developing by the binomial theorem, we have
The series terminates when x is an integer.
If, for example, we are given the initial condition that, when
x~ 0, u-= e'^y, we have
u = +
An alternative form of the solution is obtainable as follows. We
can write *(2) in the form
= {E»+l)®si(«/-a:),
whence, developing as before,
(3) u= 4>{y-x) + (^^4>{y-x+l) + (^<j>(y-x + Vj + ... .
13*81. An Alternative Method. Let us again consider the
equation
u{X'\-l, y^-l)-u[x, y+l)-u[x, y) = 0.
Assume that
u = 'LCa^h^,
where, the summation extends to an unspecified range of values
of a and 6. Substituting, we have
S (a6 - 6 - 1) C a® 6^ = 0,
so that the postulated form is a solution, provided that
a6 “ 6 - 1 = 0,
which gives a == (1 + 6) 6““^ and we have the solution
u = SC(l + 6)“=6i'“®.
426 THE LINEAR DIFFERENCE EQUATION [13-81
Since C is perfectly arbitrary, we may replace C by where
(j) (b) is an arbitrary function, and the summation may be replaced
by an integration. Thus we have
^ _ f Jyy~x ^ J ^
J —a
In this expression <[> {b) being perfectly arbitrary may be taken
to vanish outside any specified interval of b, so that we can take
for limits of integration any pair of arbitrarily assigned numbers
and still obtain a formal solution. If we expand (1 and then
write
ir{z)=:r^b^<l>{b)db,
J -00
we obtain
“ = (2/ - ^^) + ( 1 ) V' (2/ - * + 1 ) + (2) V" (y - + 2) + ■ . •
which agrees with 13-8 (3).
13*82. Equations Resolvable into First Order Equa-
tions. Consider the equation
2 2
y) = AyU{^, ^“1).
Replacing u{x, y) by u, we have successively
2 2
(A® E* ^ - Ak Ej/” ^)u = o,
2 2
(Acc Ey — Ay Ex) ~ ^7
(ElE.+ E.-E^Ex-Ex)^ = 0,
(ExEy-i)(Ex-Ev)^ = 0.
The last equation is resolvable into the two equations
(Ex Ey-l)t^ = 0, (Ex- EJ'i^ = 0.
The first gives
X ^
of which a solution is
= i.E.y^Y4>(y) =
13-82] WITH CONSTANT COEFFICIENTS 427
and the second, see 13*8, example 1, gives
u = \l/'{y~\-x).
Thus the general solution is
u= w^{x, y)cl>{y-x) + m^{x, y) {y -i- x) ,
where are arbitrary functions periodic in both x and y with
period unity.
13*83. Laplace’s Method. Consider the equation
Aqu{x, y) + A^u{x-l, y-l) + A^u(x-2, y-2)-i-...= V (x, y),
where Aq, A;^^, Aq, ... are independent of the variables and V(x, y)
is a given function. The characteristic property of this equation
lies in the fact that the difference of the arguments in any one
of the functions u{x~s, y-s) is invariant for 5 = 0, 1, 2, ... and
equal to x - y. Putting
x-y=: k, u{x, y) = u(x, x-k) =
the equation becomes
Aq'^^x'^ Ai ^x-l "t -^2 '^■x-2. -^5
which is an ordinary equation with constant coefficients. We solve
this and then replace k by x-y and the arbitrary periodics by
arbitrary functions of the form
w{x, y)(j>(x-y),
where m{x, y) is periodic in x and y with period unity.
Example. A and B engage in a game, each step of which consists
in one of them winning a counter from the other. At the beginning
A has X counters and B has y counters, and in each successive
step the probability of A^s winning a counter from B is p, and
therefore of B’s winning a counter from A is l-p. The game is
to terminate when either of the two has n counters. What is the
probability of A winning ?
Let u^^ y be the probability that A will win, any positive integral
values being assigned to x and y.
428 THE LINEAB DIFFEBENCE EQUATION [13-83
Now winning the game may be resolved into two alternatives,
namely,
(i) his winning the first step and afterwards winning the game, or
(ii) his losing the first step and afterwards winning the game.
Thus
V ^ !P '*^£0+1, v-1 + (1 ““ p) 1/+1*
In this equation the sum of the arguments in any particular
term k x-{-y. We therefore use Laplace’s method and put
which gives the equation
which has the solution
'1-pY
and hence
««.v = ®i(»> y)<f{^+y)+i^!t(so, y)(/Y)
In the present case the variables are positive integers, so that
the arbitrary periodics are constant and can be absorbed into the
arbitrary functions \Jr. Thus
«,» = 4>i^+y)+(^^y i'ix+y),
Jr
and we have to determine the arbitrary functions.
The number of counters h is invariable throughout the game.
Now Ak success is certain if he be ever in possession of n counters.
Hence, if x = n, = 1, and therefore
Again, A loses the game if ever he have k-n counters, for
then B has n. Hence, if x = = 0, and therefore
13*83]
WITH CONSTANT COEFFICIENTS
429
Putting
we obtain
whence
qn-~y _ I
® ~ q2n-x~v-l
_ { ??”->' - (1 - p)"-!' } jO"-®
p2n—x—y _ _ p'j2n—X'-y ’
which is the probability that A will win.
EXAMPLES XIII
Solve the difference equations :
P '^x+^ -
2. '2^05+2 4^05+1 + ^'^X —
3. 'i^a;+2 + 2z4a.4.i + Wa. = aJ(£C- 1) (a? - 2) + OJ ( - 1 )®
'^aj+2 ~ '^x+i + (W^^ + Uy. = m*.
2
A^a5 + A^a; = ^+sina;.
6* '^®+4 “ 6^a:+2 + = 3)».
3
A^aj-S A'^aj + ^'i^a; = 2®(l + cosa;).
6 6
3* A ^a;+l 2 A ^a; ™ 3/“h 3®.
9* ‘^a5+2±^^'2^!r = COS mX,
P>. ^^a.+4 ± 2^2 + ^^4 ^
IP '^aj+n-l + '^aj+n-2 "t • • • + '2^a; =
12. =
13. = {l-m)x,
'^x+i-% = (m-n)a?,
“■ ~ Z) 27.
430 the linear difference equation [ex. xni
14. = 2m(ir+l),
15. '1^2+2 2'^£c+i ■“ ^
16. Solve the equation
Wju '2^05+1 '^•^a;4-2 ~ ® (^aj "t “^aj+l "t" '^a;+2)
by assuming == tan v^.
17. Shew that the general solution of the equation of Ex. 16
is included in that of the equation w^j+s - = 0, and hence
deduce the former.
18. Solve the equation
'Iffx+X ^33+2 "I” *^33+2 "I" ^aj ^aj+1 — ^ •
19. Solve the equations
^03+1= (^~^^)^a; + ^a3>
- {2m + l)'ya. + t^aj5
and shew that if m be the integral part of Jn, converges
as X increases to the decimal part of Jn.
20. If % be a fourth proportional to a, b, c; b. fourth pro-
portional to b, c, a] and to c, a, 6 ; and if b^, depend in
the same manner on b^, q, find the linear difference equation
on which depends, and hence shew that
an — a
{be a~
21. Solve the equation
^a+5
^03+4
'33+3
^a+4
^as+3
'33+2 ~
^aj+3
^33+1
and consider in particular the case (7 = 0.
22. If t;o, ^2, ... be a sequence, the successive terms of which
are connected by the relation
Ex.xiu] WITH OONSTAHT COEFFICIENTS 431
and if Vq, % be given, prove that
— 2 cosacosmQc~t;Qsin(m- l)a
”* sin a
where Vi = 2 cos a. [Smith’s Prize].
23. If n integers be taken at random and multiplied together in
the denary scale, shew that the chance that the digit in the units
place will be 2 is
24. Shew that a solution of
^x+n ^x+n—l * * * ^x ^ {'^x+n
is included in that of t^a;+n+i “ = 0, and is consequently
+ ^2 0^^"" + • • • + ^n+1
where a is an imaginary (w + l)th root of unity, the n + 1 periodics-
being subject to an equation of condition.
25. A person finds that his professional income, which for the
first year was £a, increases in Arithmetical Progression, the common
difference being £b. He saves every year 1 / m of his income from
all sources, laying it out at the end of each year at r per cent,
per annum. What will be his income when he has been x years
in practice ?
26. The seeds of a certain plant when one year old produce
ten-fold, and when two years old and upwards produce eighteen-fold,
A seed is planted and every seed subsequently produced is planted
as soon as it is produced. Prove that the number of grains at
the end of the Tith year is
1 (/U + aY /ll-~a\^}
a{\^) ~V^)
where a = 3^17.
27. A series is formed by taking each term as the arithmetic
mean of the three terms preceding it. Shew, if be the 9^th term,
that when n is large
Un - nearly.
432 THE LINEAR DIFFERENCE EQUATION [ex. xm
28. Three vessels contain water. Of the contents of the first, 1 / ^
is transferred to the second, 1 / ? of the second is then transferred to
the third, and then 1 / r of the third is transferred to the first. The
cycle of operations is repeated many times. Shew that the fraction
of the whole volume of water which the first vessel then contains is
nearly
P
29. Two closed vessels A and B each containing gas are connected
by a sliding shutter which is opened for t seconds and then closed.
This operation is repeated a large number of times. Each time the
shutter is open 1 / a of the molecules in A penetrate into S, while
1/ & of the molecules in B penetrate into A, Initially there are
jp molecules in A, and q in B. Find the number of molecules in
each vessel after the shutter has been opened n times. Shew that
after a long time has elapsed the number of molecules in A and B
are in a fixed ratio, nearly. [Royal Naval College.]
30. A circulating library is started with b books. During each
year 5 per cent, of the number of books which were in the library
at the beginning of the year are added to it. At the end of every
third year 10 per cent, of the books are worn out and are destroyed.
Shew that at the end of n years the number of books is
6 A" [1 + CO” + + 0 (1 + 0)”+^ + co^^+^) + 0^ (1 + ] ,
where 9c® = 10, i = 21 / (20c) and o> is an imaginary cube root of
unity. [Royal Naval College.]
31. A large number of equal particles are attached at equal
intervals i'l to a massless inextensible string. The first particle is
projected vertically with velocity V and the particles start one by
one into vertical motion. Shew that the nth particle will rise from
the table if
37® > qhn{n-l) (2n - 1) .
32. A curve is such that, if a system of n straight lines, origin-
ating in a fixed point and terminating on the curve, revolve about
that point making always equal angles with each cither, their sum
WITH CONSTAOT COEFFICIENTS
433
Bz. xni]
is invariable. Shew that the polar equation of the curve is of
the form
r = a+OTjCos6 + ®2Cos26+ ... + ron_iCOS (n- 1) 6,
the fixed point being the pole. Shew in particular that the curve
-hxi- y^f = {x^ + y^)
satisfies the required condition.
33. Find the curves in which, the abscissae increasing by the
constant quantity unity, the subnormals increase in the ratio
1 / a, and shew that
y^ = 6a*+c
is such a curve.
34. Find the general equation of curves in which the chord drawn
through the origin is of constant length.
35. Find the general equation of the curve in which the product
of the two segments of a chord drawn through a fixed point shall
be invariable, and shew in particular that
r =
is such a curve, being the invariable product.
36.
37. v-¥n~
38. Ux+2 s+2 ~ 6.
39. Wx+3, 6 «+l'b6 Mx+l, V+2~^a:i V+3 ~
40. u{x+l,y+l)-au(x+l,y)-bu{x,y+l)+abu{x,y) - c®+».
41. u{x+Z,y)-ia^u{x+l,y+2)+a^u{x,y+Z) = xy.
42. The probability of a coin falling head is p. What is the pro-
bability that at some stage in n consecutive spins the number of
heads exceeds the number of tails by r ? [Burnside.]
CHAPTEE XIV
THE LINEAE DIFFEEENCE EQUATION WITH EATIONAL
COEFFICIENTS. OPBEATIONAL METHODS
After equations witli constant coefficients the linear equation
whose coefficients are rational functions of the independent variable
ranks next in order of simplicity. Boole devised a method of
symbolic operators for attacking the problem of solution of such
equations. By generalising the definitions * of these operators it
is possible to apply the method to a well-defined class of such
equations and to obtain solutions in Newton’s factorial series .in
much the same manner as power series solutions of differential
equations are obtained by the method of Frobenius.
14'0. The Operator p. Given a fixed number r and- an arbi-
trary number m, the operator p is defined by the relation
This is a generalisation of the definition given by Boole which
corresponds to the case r = 0, m an integer. In particular, for
m = 1,0,-!,- we have
pM(x) = {x~r)u[x-\),
p® u{x) = u {x),
p~^ u (x) =
f^u{x) =
1
x-r+1
r(a;-r-H)
r{a:-r-4-f)
u{x+l).
*L. M. Milne-Thomson, On Boole’s operational solution of linear finite
difference equations, Proc. Cambridge PUl. Soc. xxviii (1932), p. 311.
434
14-0] BATIONAL COEFFICIENTS. OPEEATIONAL METHODS 436
Ifj for brevity, we write
(2) x-r = ^,
the above definition becomes
The operator is clearly distributive. That the index law is
obeyed is easily seen, for
r(^^+i) p-.. f r(x-+i) ,
r(cc'-m+ 1) lr(ic' -
;(£r-n)|
■ u{x-'m-n) = pw+n ^ (^x) .
r{x' -m-n + l)
If the operand be unity, we shall omit it and write
^ r{x' -m+l)
Hence, when w is a positive integer, we have
pM+n = (x' -m){x' -m-1) -m-n+l)T{x'-{-l) ir{x' -rn+l),
Qin-n — i ^ ^
^ {x' -m-\-l){x' ~7n-\-2) ... {x' -7n+n) V{x' ’
It follows that a series of the form
is equivalent to
r(a:' + l) ( fx' -m\ , fx'-m\ ,
TV^dT) l^"V n
Kl\
A-CIq-A-t-
6o2!
® ‘ x' -m+l ' {x' -m+l){x' -m+2)
while a series of the form
+ ...
m+l
+|fp’"+=+-
436 THE LHSTEAR DIFFERENCE EQUATION WITH [U-0
is equivalent to
r(x' + l)
( fx' -m\ . fx' -m\ , /x'-m\ )
|«o+«i( 1 2 3 ;+•••}'
In this way we can, apart from the Gamma functions, express
factorial series as series of powers of p operating on unity. Con-
versely, a series of powers of p can be interpreted by means of the
above results.
14*01 . The Operator tc. The definition of n is
rcu(x) = = x' {u{x)-u{x- 1)).
-1
Boole’s definition corresponds to r = 0. It should be particularly
noted that just as a fix:ed number r is associated with the operator p,
so we associate the same number with the operator tz in all cases
where re and p both occur in the same work.
The operator n is distributive and can be repeated. Thus we can
interpret u (a?), tc" u {x) where n is a positive integer. The index
law is clearly obeyed, and n commutes with constants. Thus, if /(X)
be a polynomial, the operation /{n) has a perfectly determinate
meaning. Moreover
/(tc) g (tc) u{x)=g {tc)/(7c) u {x) ,
where/(X) and g (X) are any polynomials.
We can now prove that, if ^ be a positive integer,
tt” P'^u(x) = p”^(Tc-l-m)’^ u (a;).
We have from the definitions
(a;) — x‘
,/ r(x^+i) .
m)-
r (x' - m)
u {x-
m-
D)
Tjx'^l)
{x' u{x-m)- {x' --m)u{x-m-l)}
r(a;^ + l)
{(aj' + m)u{x)-x' u{x-l)}
= p^ (T:+m)u{x),
14-01] RATIONAL COEFFICIENTS. OPERATIONAL METHODS 437
Kepeating the operation,
= 7r[p”*(7T+m) = p’^(7r + m)[(Tc + w)^/(a;)]
= p^(7T+m)^ w(a;),
and continuing thus, the required result is proved for any positive
integral index n.
From this we can infer the more general result that, if /(X) be a
polynomial,
/(tt) ^‘^u(x) = p’"/(7c4-m) w(a;).
14*02. Inverse Operations with tt. The equation
nu{x) — <j>{x)
has the symbolic solution u{x) ^ ^{x) and also the particular
solution
z
c
SO that a possible interpretation of is given by
7r"*^^(x) =
^6(t)Ai+C,
0— r _ 1
which gives
7zt:~^(I>{x) = (l>(x).
If we use this interpretation, we have, from 8*1(6),
X
Tz~^-n:<l>{x) = ^ A Ht) A t + C= ^{x) + K,
where Z is a constant whose value depends on the particular
value attributed to C. We shall suppose 0 to be so determined
that K is zero. If this be done, n and are commutative
operations.
We therefore make the following definition :
z
(1)
n-^cf,[x)= Q L^(i)At + C,
0-r „x
438 THE LINEAR DIFFERENCE EQUATION WITH [14-02
where the constant C is given a particular value which makes
rc<j>[x) =
In practice it will seldom be necessary to determine (7, but the
above definition enables us to attach a precise meaning to and
thereby gives a wider range of application to this operation.
We can now interpret tc” when n is a positive or negative integer,
and we have for all integral values of p and n
7U^ (x) rp cf) (x).
To interpret (rc + m)"^ u{x), we have, from 14-01,
(7u + m) p“’^[tc“^ p^u{x)]z^ P“^7U7U“^P"^w(x) — u{x).
Thus a possible interpretation of (tc + m)’"^ u {x) is
(2) (tt + m)"^ u {x) ■= p^" u (x)y
and if we adopt this we have just proved that
(7T:+m)(Tc + m)'^^u(x) = u(x).
Moreover, (2) gives
(tc + m)“^ (tt + m) w (a;) = p"’” p^[(7z-hm)u{x)]
= p~w ^-1 ji: p^u(x) = u (x).
Thus, with the interpretation (2), we have
(7v-i~m)~'^(n-hm)u(x) = (7i: + m)(n-i-m)-^ u(x) = u(x),
so that the interpretation (2) makes the operators Tc + m, +
commute, and this interpretation is therefore suitable inasmuch as
it preserves the commutative property of k and tc“^ when m = 0.
It follows at once that, when p and n are integers,
(tc +■ m)^ (tc + m)” u = (tt + u.
Prom the commutative property of the direct and inverse opera-
tions we have the important result that if /(X), ff (X) be two rational
functions of X, then
f{7c)giTt)u=g{Tz)f(n)u,
and further that, to interpret /(tc) u{x), we may express /(X) as
the sum of terms of the form jB(X-6)^ (that is, we may
use the method of partial fractions). Thus we see that /(tc) u{x)
is the sum of terms like A tz^u (x), JS (tc - 6)^ u (x).
14 03] RATIONAL COEFFICIENTS. OPERATIONAL METHODS 439
14-03. The Operators tc^ and These are defined by the
relations
7Ziu{x) = x' ^u{x) = x' {u{x-\-l)-u(x)},
p^u{x) = E’”w(a;), x' = x-r.
With the necessary modifications in the definitions the results
which we have already obtained for the operators iz and p apply
to the operators and pj, and it will be unnecessary to repeat the
arguments.
Further generalisation can be made by substituting A A
the definition of tu, and a corresponding change in the definition of p.
The cases which we consider are those in which (o = l or6) = -l.
We shall now proceed to prove some general theorems for the
four operators tu, p, Tr^, pj. The theorems will be stated for both
sets of operators, but will be proved only for tt and p. The reader
will have no diflS.culty in supplying the proofs for and p^.
14-1. Theorem I. ///(X) be a rational function, then
/{n) p"^ u p^/(tc + m) u,
/(%) Pf M= pf/(Ki + m)M.
The theorem has already been proved for the case of a polynomial
(see 14-01).
Suppose /(X) expressed in partial fractions, say,
where P(X) is a polynomial. From 14-01,
(tt -■ a)^ p^ [(tt — a i- m)~^ u] = p’” (tt - a -f m)^ [(tu - a + m)"” u] = p'^u.
Operating with (7r -a)~”, we have
(TT-a)"” = p^(7r-a + m)“ ^u.
The theorem is therefore true for /(X) = and is there-
fore true for any rational function.
440 THE LINEAR DIFFERENCE EQUATION WITH [U-11
14*11. Theorem il.
7t(7T- 1)(7U-~2) ... (7T-n+l)t^ = x' {x' -l){x' -2) ... (x'-n+l) A-m
- l)(7Ti-~2)...(7Ti~n+l)w = a;'(a;'4-l)(a;' + 2)...(a;' + n~l) A^^.
From Theorem I, we have
(tC - /c) = (tT — Ic) p”* W ::p p* TC p”* U,
SO that
{7i:“nH-l)(7t-“n-f 2) ... (7i-l)7r w
rb p^~^ TT p~”+l p’^-"^ ^ p-n+2 p-2 p ^ p-1 ^ ^
p”(p“‘^ 7U)”t4.
Now
Tct4(a?) = x' {u{x)-u{x- 1) ],
and therefore
p-^Tcw(x) = (aj' + l)[te{a;-|-l)--ii(a;) ] -f- (a;' + l)
= A u{x+l) = E A
-1 -1
Hence
{p-i7r)"M(a:)= E” A u{x),
-1
pn(p-l^)n^(2;) x' {x' -1) ... {x' -n+l) E”” E” A
SO that
n
(jc-.^+l)(7c-n+2) ... (tu- 1) Tcw = a;'(cc'-“ 1) ... (cr'-n+l) A
-1
which is the required result.
14*12. Theorem 111. If F{k) be a polynomial,
J (tT + P) M i [j’(k) + (7.) p + i (tt) ^ ^3 (tt) p3 + . . . "
P,) w = [j (tci)- ^.(Tri) p? +
u,
where
u,
J^.W=AF(X), n = 1,2,3,....
U-12] BATIONAI. COEFFICIENTS. OPERATIONAL METHODS 441
The theorem is clearly true for
(1)
Suppose it to be true for
(2) F{X)=:
that is to say,
since
'tc+p
I
l-s
\n~s.
u,
Operating with (n + p-n) / {n-i-1), we obtain
/7T+p^
\n+:
+
TT-S+l 1 1
u.
From Theorem I, the second term in the square brackets is seen
to be
i-s + lJ (5-1)!’
so that we at once obtain
The theorem thus follows by induction, from (1), when F(X) is
of the form (2). Since any polynomial can be expressed as the
sum of terms of the form (2), the theorem is proved.
The application of this theorem is as follows :
We have, from the definitions,
=:{x-r) {u^ - pu^:=z(x~T) ,
whence, by addition,
(Tr+p)^ =
Thus multiplication by cc' or aj-r is equivalent to operation with
TT + p . Symbolically,
z' = x-rz=7t + pj
442 THE LINEAR DIFFERENCE EQUATION WITH [14-12
We can now express any pol3rnomial as an operator for, if f{x)
be a polynomial,
f{x)U:=f{K+() + r) u,
and v/e can api Theorem III to developing the right-hand member.
Thus, for example,
= (Tc-fp-l-r)“ u.
Here
F(X) = {X+r)3
= (X-l-r)(X-f-r— l)(X-fr — 2) + 3(X-l-r)(X + r— l)-!-(X-l-r),
i?’i(X) = 3(X + r-l)(X + r-2)-f6(X + r-l)-Hl,
f2(X) = 6(X+r-2)-l-6, 2?3W = 6.
Thus
a:5M = {(-n:+r)^-f-[3(7r+r)®-3(7r + r)H-l ] p-l-3(TC + r- 1) p2-|-p®}M.
14 'IS. Theorem IV. //'/(X) be a rational function, then
/(u)p’».l=/(m)p“l,
/K)pf-1=/Mpri-
By Theorem I,
/(u) p™! — p”*/(T:-Pm) 1.
By Taylor’s theorem, we have
f{m)+f\m)-k+ f' {rn)~+ ... ,
and therefore
/(Tc+m) 1 =/(m) l+/'(m) tu . 1 + ... .
But TT . 1 = 0, 71^ . 1 = 0, and so on, so that the theorem is true
for a polynomial.
To prove the theorem for any rational function, since we may
express the function in partial fractions, it is only necessary to
consider the case /(X) = (XH-a)“". By the iSirst part,
(TT+a)^ ~ pw I
Operate with (m + a)"^ (tt + a)-", and we have
Thus the theorem is true for any rational function.
14-14] RATIONAL COEFFICIENTS. OPERATIONAL METHODS 443
14*14. Theorem V. Every linear difference equation whose
coefficients are rational functions of x can be expressed in either of
the forms
[5'oK)+fl^l("l) Pi +5^2 W Pl + pf] u{x) = g{x),
where fs{'k), g^ih), (s = 0, 1, 2, , m) are polynomials and f{x),
g{x) are known functions of X.
Suppose the equation to be given in the form
(1) X^u{x)-^X-^u{x~l)-^X^u{x-2)-\- .,,-^X,,u[x~n) = X,
where the coejBBLcients are rational functions of x, which may, without
loss of generality, be supposed polynomials, since multiplication of
the equation by a suitable polynomial will produce this case.
Multiply the equation by x' {x' -l){x' - 2) ... {x' -n+l) and
observe that x'u{x-l) ■= ^u{x), x' {x' -l)u{x-2) = i^^u(x), and
so on. The equation then assumes the form
<f>0 (x) u (x) + (x) p u(x) + ... +cf>„(x) p" U ix) = f{x),
where the coefficients are polynomials in x.
Now we have seen that multiplication by x' is the equivalent of
operation with 7u + p, and therefore that multiplication by x is
equivalent to the operation tc + p + r. If then we replace x in the
coefficients by tc + p + r and expand these coefficients by using
Theorem III, we obtain the form stated in the enunciation.
If the equation be given in the form
(2) u{x) + 1) + ... -hX„ u{x+n) = X,
we can multiply by x' {x' i-l) ...{x' + n-l) and put
x' u{x + l) = pl^^(^c), x' {x' + l)u{x-{-2) = ^iu{x),
and so on. If we then replace x in the coefficients by - ttiH- Pi + r
and again use Theorem III, we have the second of the forms stated.
Since equation (1) can be transformed into the form (2) and
vice versa, each of these equations can be expressed in either of
the forms given in the enunciation.
We shall call the forms given in the enuncia.tion the first and
second canonical forms respectively.
444 THE LINEAR DIFFERENCE EQUATION WITH [U-U
The above theorem is fundamental in Boole’s method and gives
rise to the following remarks :
If the equation be given in the form
^0 A + a u(x) + ,,• -j- J[j^u(x) = X.,
-1 -1
the simplest procedure is often, not to reduce it to one of the forms
used in the proof of Theorem V, but to multiply by
x' (x' - V){x' -2) ... {x' -n-tl).
Then, by Theorem II, the equation assumes the form
<1>q{x) 7c(7u~-1) ... {7c-n+l)u{x)
+ <l)i{x)n{To-l) ... {'K-n + 2)u{x)-\- .., = f{x).
If we then replace x in the coeJBhcients by Tr + p + r and effect
the proper reductions by use of Theorem I, we arrive at the first
canonical form of Theorem V.
It might also be noticed that another way of reducing the equation
XqU{x)'^X^u{x- l)^.,,-{‘XnU{x-n) = X
is to make the change of variable
u{x) = v(cc)/r(x' + l),
which gives
and so on, so that we obtain
ZoU(a5) + Zip?;(a:)4-...+Z„p"u(a:) = Zr{a;'+1),
which, can be reduced to the first canonical form by the method
already explained.
In the same way the equation
Xq u (x) h- (a; + 1) 4- . . . + X^ u^x-j-n) = X
becomes by the substitution u{x) = v(x) r(a;'),
Zo « (a;) + Zi Pi i; (ir) + . . . + Z„ pXa;) = Z / r (a:' ) .
14'14] KATIONAL COEFFICIENTS. OPEBATIONAL METHODS 445
It must, however, be clearly understood that a change of the
dependent variable of the kind just described may so affect the
solutions of the transformed equation as to render the method of
solution in series which will presently be described inoperative. On
the other hand, should an equation when reduced by Theorem V
prove intractable, the change of variable may lead to an equation to
which our method will apply.
14‘2. Formal Solution in Series. Consider the homo-
geneous equation
(1) .Xfl u(x) + Jfj M (a; — 1) + ... +^„u(x — n) = 0,
where the coefldcients are polynomials. We first make the change of
variable u(x) = The equation then becomes
+ 1)-|- ...
+ (x-n+l)+X„v (x-n) = 0.
If this equation be reduced to the first canonical form, we have
(2) [foM + A(^)P + f2Mp^-<-"- + fm(T^)p”']v(x) = 0,
where /o(7r), fi(n), , fmin) involve the parameter p rationally.
For the moment we shall leave (x imdeterroined, and we seek to
satisfy the equation for ■u(a;) by a series of the form
(3) v(x) = + + ,
where the operand unity is understood. Substituting this series in
(2), we shall have a formal solution of the equation if the coelBicients
of the several powers of p vanish. Using Theorem IV, we thus
obtain
(4) ao f^(m+ k) = 0,
% - 1) + “o /m-i(“+ 1) = 0.
®2/m(w + lb-2)-f-Oi/„_i(m+A:-2) + ao/„,_2(w-[-A:-2) = 0,
(5) a,f„(m+lk-s) + o,_i (m + i - s) -h . . .
+«^/o(«»+^-s) = 0 (s>m).
If we suppose aQ 0, equation (4) yields a certain number of
values of k, say, k^, , ky, which for the present we shall
446 THE LINEAB DIFFERENCE EQUATION WITH [14-2
suppose to be all different, and snob that no two of them differ by
an integer. The equation
(6) /™(OT+i) = 0
will be called the indicial equation.
To each root of the indicial equation there corresponds a series of
the form (3), whose coefficients are determined successively by the
above recurrence relations, which can be successively evaluated,
since, by hypothesis, no two roots of (6) differ by an integer, and
therefore, if it be a root of (6), /„,(m+^-s) ^ 0, s = 1, 2, 3
Each series obtained in this way is a formal solution of (2).
Denote the solutions corresponding to Ic^, , K by
We have then the formal solutions
of (1). Whether these solutions converge can of course be examined
in any particular case. Whether they are linearly independent is as
yet undecided.
If V = we have obtained n solutions, but if v < n the equation
has other solutions which we have yet to determine. Leaving
these questions for the present, it may happen that the indicial
equation does not contain m+A, in other words, that /„i(tc) is
independent of n. If (tt) be also independent of pi the method
fails completely, but if /^(Tr), while independent of tt, be not
independent of pi we choose, if possible, a non-zero value of pi
such that vanishes. Let pi^, pig, ..., px be the distinct non-
zero values of pi which cause /^(Tr) to vanish.
To each such value of pi we have an equation of the form
[fo{^)+fi{'^) P+-+/m-i{Ti:)p”‘-^]v(a;) = 0,
and we attempt to satisfy this equation by a series of the form (3).
If corresponding to pi = pq this equation yields formal solutions
v^{x), .... (x)j we have as solutions of (1)
iifvjx).
Similarly for pig, p.3, ..., pi^ we may obtain corresponding sets of
solutions.
14-2] BATIONAL COEFI'ICIENTS. OPEEATIONAL METHODS 447
Thus we see that if equation (1) have a particular solution of the
form
P*' + ai p'^~^ + «2 p®“^+ ••• },
this solution will in general be detected by the above method. Since
_ r(a:' + l) _ r(a:-r+l)
P r(a;'-m+l)~r(;r-r-m+l)’
we shall expect our method to determine any solution of the form
^^r(a;-r4-l)/^ 1
r(iC“A-r+l) I ^ {x-T-h-Vl) {x-r-h^l){x-r-k-^2)
Example. {x-2)u{x)~~{2x-2>)u{x-l)-?>{x-l) u(x--2) = 0.
Putting u{x) = \x^v{x), we have
(jL2(a;-2) v(a;)-- pL(2a;-3)i;(a;- 1)- 3(a;~ l)v{x-2) = 0.
If we take r = 0, so that
xv{x- 1) “ ^v{x), x(x-l)v{x-2) = p^v{x),
the equation becomes, on multiplication by x,
[\i^{x-2) X- (i.(2a:-3) p--3
Writing 7t4- p for x, we get, by Theorem III,
[[x2{ 71^ ”271:+ (271 -3) p + p2}- [ji(27i:“3 + 2p)p-3 p^]v{x) = 0,
[^2(7^2_27r) + ([jL^- pt)(27T:-3) p + ((x2”2p-3) p^]'y{a?) = 0.
Since = p.^-2p~3 is independent of tu, we choose p so
that p2”2[ji-3 = 0. This gives [x = 3 or -1. With either of
these values for p, the equation becomes
[p(7r^- 27r) + (p- 1)(27t- 3) p]t?(a;) = 0.
Assume
v{x) = (Zo p^ + aip*”^ + a2 p^”^+....
The indicial equation is
2 (^^4- 1) - 3 = 0, whence k = -J.
The recurrence relation for the coefl5.cients is
((ji”l)(2i: + 2-25”3) a5 + p(£+l”5)(t-l -s) = 0,
THE LINEAR DIEEERENCE EQUATION WITH
[14-2
448
whicli gives, since h = \.
(2s- 3) (2s + 1)
Os = ? ^ o.-i, P =
o 23-5
Oil — ” O2 — p 2 ^0? ^^3
8(t4-l)’
,3. 3. 5. 7
OJ
Os = -i’*Oo[(2^-3)(2s-5)...6.3(2s + l)(2s-l)...3]/s!.
When [1 = 3, p = fV, and when [x = - 1, p =
Hence we have the formal solutions
,, 3“=r(a:+l)f, 3.3 3^ 3.5 1
«iW- r(a:+|) I 16(ic+^) 1# (a:+i)(a; + |)2 ■■■/’
^ h _ -A _ I
r(x + i) r 16(^ + i) 162(x + -|)(a:+-«)2 -J’
where the successive values of the coefficients are determined by
the formula above for a^.
If ts denote successive terms of either series,
^ y(2^~3) (2^+1) ^ ^ (i+i.o
Thus, for %(cc), 7^ and for 1^2 W 5
Thus both series are absolutely convergent.
14*21. Solution In Newton's Series. The method of opera-
tional solution can also be applied to finding a solution in the form
(a;' - ^) (a:' - ib - 1) . .. (a;' - ^ - s + 1) .
The equation having been reduced, after the substitution
u {x) = p.® -y (x) ,
to the form
[/o(’T^) + /iWp+ — + /fl.(w) p’”]n(a:) = 0,
and (X chosen, if possible, so as to make the term fm{v:) vanish, we
substitute
v{x) = Uop^ + Oi ?*=+>•+. .. + a,p*=+’+... .
14-21] RATIONAL COEFFICIENTS. OPERATIONAL METHODS 449
Equating to zero the several powers of p, we obtain the indicial
equation
/oW = 0,
together with the recurrence relations
+ + + + + ••• = 0,
by means of which the coefficients can be successively determined
when the value of Tc has been found from the indicial equation.
If the series obtained in this way be convergent, we have a
solution of the equation in the required form. The following example
illustrates the method :
Example.
{x~-ol){x-~ i^)u{x)-[2x{z l)-8{x-l) + oii^]u{x-l)
■\-{x-2){x-y-l)u{x-2) = 0,
where
S = a+ (3 + y+I.
Take x' — x-l. Putting u{x) = il^v(x) and multiplying by
x-l, we get
(tt + p) (tt + 1 - oc + p) (tt + 1 - p 4- p) -y (a?)
- iJL[2(7r+p)(7r4-l + p)~S(Tc+p)+a (3] p'y(cc)
+ (tt + p -y)p2'y(a;) = 0.
The coefficient of p^ is ([i - 1)^. We therefore take p = 1, so that
u{x) = V {x), and the equation reduces to
[■n:(7r“a+l)(7r- ^4- l) + 7u (Tr-a- (B + y) p] w (a?) = 0.
Put
u{x) = Cq p^4-aip^+i4-...4-asp*+®4-... .
The indicial equation is then
^;(A;-(x4-1)(&- (B4-1) = 0,
whence
= 0, a— 1, |3 “ 1.
The recurrence relation for the coefficients is
(^4~5 “• a4“ 1) (^4" 5 ■“ p4~ 1) 4- (it 4- s) (it 4- 5 — a — [B 4- y) = 0,
450 THE LINEAR DIFFERENCE EQUATION WITH [14-21
SO that we have for a, the expressiori
(-l)‘'(jS;+l-a-fi+Y)(A;+2-(x-p+Y)--- a- p+y) an
(A+2-a)(fc+3-a) ... (jfc+s + l-a)(A+2-j3)(A+3-p) ... (A + s+1-^ ‘
Putting in turn A = 0jii: = a — 1, ib = p- l, we have the three
solutions
Wi(a;) = 1 +
(-l)®(l-a-p + Y)...(s-a-j3 + Y)5!
,tl(2-a)...(s+l-a)(2-p)...(s + l-p) V s J’
u^{x) =
r(x)
r(aj-a+l)
(-1)°(Y-P)---(y-P + s-1)
(a- 1) ... (a- p + s)
« _ r(a;) r f (-ip(Y-a)-(Y-°c+^-l)
“3^^^“r(ir-j3+l)L ^ (p-a+l)...(p-a+s)
The ratio of the 5th term to the preceding term is
(jb+g-Qc-- ^4-y) _ 2 - Y + 2
{i+5-“a+l){^fc+5- p+1) 5
+ ... ,
All three series are therefore absolutely convergent if
R{x)>R{y-l).
In the case of neither a nor p may be a positive integer
greater than 1.
For U2(x), a- p must not be a negative integer, and in the case
of u^{x), p-a must not be a negative integer.
These three solutions cannot be hnearly independent.
If a = p = 1, the indicial equation has a multiple root and the
three solutions coincide. The method of dealing with multiple roots
of the indicial equation is discussed in section 14*22.
This equation can also be satisfied by factorial series of the first
kind, for, putting
u{x) = p^-®+...,
we have the indicial equation
(A-f l)(^+l-a-- p + y) ==
whence Jc=-l or a+^-Y""l*
The recurrence relation is
i^g(A-54-l-a-p + Y) + ^^5-i(^“-5 + 2-a) (/b~5 + 2- (3) = 0,
14-21] RATIONAL COEFFICIENTS. OPERATIONAL METHODS 451
wMcli gives for b, the expression
(g-2-ii:+a) ••• ( - 1 - ^+«) (s-2-il;+ P) ... ( - l-/b+|3)
■ (s-fc-l + a+ 13- Y).(s-^;-2 + a+ p-y) ... (-A+a+ p- y) '
Writing, for brevity, c = a + p- y+l, we have the solutions
( \ I «(«+l)P(P-f 1)
cx{x-vl) e(c-¥l)x(x+l)(x^2)
a(a+l)(a4-2)p(p + l)(p-f2)
'^o(c+l)(c + 2)a;(a;-t-l)(a;-t-2)(a;+3)*^ ’
uAx)
T(a:
Tjx) r, , (y-«)(y-p)
c-c+2) L l!(»-c+2)
'4- (y-«)(y-«+l)(y-P)(y-p + 1 )
^ 2!(a:-c + 2)(a;-c-f3)’
SO that in terms of the hypergeometric function
+
Mg (a:)
r(x)
-F{y-x, y-p; x-c-l-2; 1).
r(x-c + 2)
Thus, using 9-82, we have
r(x)r(x-y+l)
“5(^)-r(x-p-fi)r{x-«+i)
The series for u^{x), u^{x) converge for R(x) > R{j- 1). When
I a; I 00 in the half-plane of convergence we have the asymptotic
relations
^4(0;) ^ u^{x)
the latter result following from 1043.
It follows, from 12*16, that u^{x), u^{x) form a fundamental
system of solutions.
14*22. Exceptional Cases. In the preceding discussion we
excepted the cases in which the indicial equation has multiple roots
or roots which differ by an integer. In the case of a multiple
root the method only gives one series corresponding to that root,
while in the case where two roots differ by an integer, the
equations 14*2 (5) may lead to infinite coefficients, owing to the
possible vanishing of for certain values of s. To
discuss these cases we shall suppose that the given equation
XQu{x)-hXj^u{x-l) + .,. + Xny'{x-n) = 0
452 THE LINEAR DIFFERENCE EQUATION WITH [14-22
has been transformed by the substitution u{x) = and that
the value of has been assigned in the manner previously described,
so that for the value of p in question the equation assumes the first
canonical form
(1) [/o('n:) + /i(Tt)p + /2(7^)P®+--- + /m(’T:)p™]«(®) = 0.
The indicial equation is
fm{m + k) = 0.
If the indicial equation have roots which differ by zero or an
integer, we begin by arranging all the roots in groups such that
any pair of roots of the same group differ by zero or an integer.
The roots of such a group will be called congruent. Let p be the
greatest positive integer by which a pair of roots of the indicial
equation differ.
Consider the non-homogeneous equation
(2) [/o (n:) + A (7t) p + . . . + fm {n) {x) = a{k)f^{m+k)
where
a(/c) = bj^{m+k-l)f^{m+k-2)...f„{m+k-p).
If round each of the roots of the indicial equation we describe
circles of radius y, we can make y so small that when k
varies in the domain K formed by these circles the function
s>p, does not vanish at all. Under these condi-
tions we can find a formal solution of (2) by putting
v{x) = a {k) + 62 4- 62 4- . . . .
Tor if we substitute this series in (2) we see that the coefficients
of are equal, while the coefficients 62, 635 ••• j &«> ••• arp found
from the recurrence relations
+ + + = 0,
&2/tn(^+^“2)4-6i/^_i(m+*-2)4-a(i)/„,_2(^+^-2) = 0,
+ K~1 /m-l(w^4- A - p) 4- .. . = 0,
+ /m^l (m 4- A - 5 ) 4" . . . = 0,
14-22] RATIONAL COEFFICIENTS. OPERATIONAL METHODS 453
and these equations lead to determinate values of the coefficients,
for should any of the coefficients + 5 = 1, 2,
vanish, we can first remove the vanishing coefficient which also
appears in a{h) and therefore in 63, Jg, ••• .
Moreover, + ^'>'Py cannot vanish for any value of
h in K. We thus obtain a formal solution of (2) in the form
a(fc) + ;
k-\-l 2)
+ .
If the factorial series
TV-F+iy
z+l~^(zi-l){z + 2) (z-hl)(z + 2){z^3y‘’'
converge uniformly with respect to 2; for R {z) > X, the series
for Vjc(x) will converge uniformly with respect to Jc for R{x) > X',
where X' > X depends on the exact disposition of the region K,
Consequently, for jR(a')>X' and k in K we can differentiate
the factorial series term by term, and we thus obtain the result
that 3^ Vk (x) I dk^ exists as an analytic function and satisfies the
difference equation
(3) [/o('n:) + "- + /™(^) ^ |a(^:)/„(m + A:) p™+*j .
Now consider
a{k)f^(m + k).
Let ao, a^, a2, ... , a2_i be the congruent roots which constitute
the first group of the roots of the indicial equation and suppose
them arranged in non-descending order of their real parts, so that
J?(ao) ^ R{<^i) ^ ^ ^
Let a^j, a^, ... , a,, denote those roots which are distinct.
Then
^0 = ai = a2=...= a;,_i,
so that is of multiplicity X.
Again,
so that ax is of multiplicity p - X, and so on. Thus
a{k) = 6o(^Sj-ax)^(A--a^)^.*. (A-OYW.
where f{k) does not vanish for any root of this group.
454 THE LINEAR DIFFERENCE EQUATION WITH [14-22
Now
Thus
where j>{k) does not vanish for any root of the group.
It follows that
vanishes when
Tc — olq, 5 = 0, 1, 2, ... , X- 1,
k =: (Xx, s = X, X+1, ... , pi~ 1,
k = 5 = V, v + 1, ... , Z~ 1,
and in these cases the equation (3) coincides with (1) and therefore
we have I solutions of (1), namely, ^[^’&(^)] corresponding to
the above set of values of k and 5.
That these solutions are linearly distinct will be proved later from
a consideration of their asymptotic values. The corresponding
solutions of the equation in u{x) are obtained by multiplying the
gs
values of -^^[Vk{x)] by
Example 1.
(a:;34‘l)^(aj) “ {2x-l){x- l){x-a) u{x-l)
-\-{x-2){x-a){x-a-l)u{x-2) = 0.
Writing u{x) = v{x) and taking x' x- a, we have
[(7u + a+p)3 + l]'y(a:j)-~ (x[2(7i: + a+p)^“3(7r + a+p) + l] p
-P (7r + a-2 + p) = 0.
The coefficient of is [jL2-2fr+l. We therefore take pi = 1,
and the equation reduces to
[(7i:+a)^ + l + (Tc + a)2 p]^(^) =
u{x) = ao p^ + a^ p^-i+...
Putting
14-22] RATIONAL COEFFICIENTS. OPERATIONAL METHODS 455
we have the indicial equation
(ib+a+l)2 = 0.
We therefore consider the equation
[(7c*fa)®+l + (7r + a)2 p]^(aj) = +
which gives
{k+-afai-\-aQ{l-{-{h + af) = 0,
(fc + a~s+l)2as + a3„i(l-f (/c + a-5 + l)3) = 0.
Writing I = Ic + a + l,
” {s-l)^{s-l-lf{s-l~2)K,. (1-Z)2 ^0-
Putting ]c = ~a-l, that is, Z = 0, we see that = 0 and
therefore = 0, s = 2, 3, 4, ... . One solution is therefore
%(^) = P“®"^ = r(a5-a+l) / r(a; + 2).
A second solution is the value when Z = 0 of
p'"'+-+p'^-+p"^+-
and from the form of we see that when I = 0,
Sa,_(s3-l)...(23-l)(-3).
di s2(s-l)2...22 "■
Thus a second solution is
M - ^ (a^-^+l)^(^+2) _ (.3_i).,^(23_i) r(x-a+l)
r(a; + 2) 5^5- 1)2... 22 T(x + s-r2)‘
The series converges if R{x) > 0.
The following example illustrates the application of the method
to solutions in factorial series of the second kind.
Example 2.
(a;“l)w(ic)-(2a;~l)t^(a;~l) + (a/-l)w(a;-2) = 0.
Multiply by x and take x' = x. The equation becomes
[x{x- 1)- (205- 1) p + == 0,
which reduces at once to
[Tc(7r- 1)~ p] u{x) = 0.
456 THE LINEAR DIFFERENCE EQUATION WITH [14*22
Clearly this has no solutions by factorial series of the first kind.
We therefore put
u (x) = aQ + % p*+^ + + —
The indicial equation has the roots 0 and 1 which differ by an
integer.
We therefore consider the non-homogeneous equation
[7r(TC~ 1) - p] = k(lc- 1) a^ik)
where
=/o(^+^)^o = k{k+l)bQ.
Putting
u{x) == aQ{h) p^ + 6i p^+^ + &2 ,
we obtain the recurrence relations
{k'\-l)khi = aQ{k) — 6o/c(^+l),
(^+2)(A;+1)&2 = 6i,
{k+s){Jc + s-l)hs =
whence
&i=
(A+5)(^;-+-5- i) ... (>fc + 2)(/c + s- 1) ... (/c+l)‘
Thus
u{x)
r(a;+l)
{lc{k + l)
+s
51
{k+s) ... (fc + 2) . (4 + 5-1) ... (4 + 1)
The series in the bracket converges in the whole plane so that u (a?)
is a meromorphic function with poles at the poles of r(a:+ 1).
We get one solution by putting 4 — 0, whence
“■‘‘'’'Sf+iO-
To obtain a second solution we differentiate u{x) with respect
to 4 and then put 4 = 0. Now
^ ^(^+i)r(cg+i) r(^+2)r(Z;+i)r(cc+i)
r{x-k+i) ^^^T{k+s+i)r{k+s)r{x-k-s+i)-
14*22] BATIONAL COEFFICIENTS. OPERATIONAL METHODS 457
Writing for tlie term under the summation sign and taking the
logarithmic differential coefficient with respect to k, we have
^(/c+2) + ^(A:+l)-^(ii;+s+l)
Putting ib = 0, we obtain the solution
x{-<i^{x~s-\-l)-'^{s)~^{s+l) + '^ (2) + (1) }.
14*3. Asymptotic Forms of the Solutions. We have
found that when the indicial equation presents roots which differ by
zero or an integer, the solutions are obtained by differentiating
partially with respect to k the expression
r(jr~r+l)
where
Wj,{x) = a{k) +
Vj,{x) = Wf,{x)
h
T(:x-~r-k^iy
x-r -k+l'^ x-r- k-^l)(;x~r-
a[k) = hQ{k~ay)>^[k~ay)^ ... (k-ai,yf{k).
k + 2)
+ .
Now, by Leibniz’ theorem, we have
d'‘'Vj^{x) ___ r(a;~r + l) d'''~Uvi,(x)
Bt ~ V t) d¥ T[x-r-k-y\) ’
and by 10*43,
r(x-r-^Ti)
QQ(a;-r+ 1)
+ Oi(:r-r f l)log^~~^| + ... + (l + f2i(a?))(log^3^
where Qq, ... , are inverse factorial series without a constant
term.
It follows that for large values of | iJ? | we can replace the right-
hand member by its largest term, namely, [log {1 / (a: - r + 1) } ]*.
Thus we have
gS-t 0^
d^v„{x)
458 THE LINEAR DIFFERENCE EQUATION WITH [i4-3
For the X roots equal to we have therefore, if we retain only
the largest term on the right,
/a»Os,(a:)\
(a: - r + 1)*“ (log « («o)
(log^) a(ao), s = 0, 1, 2, ... , X-1.
For the roots equal to we observe that is a root of
i{Jc) = 0, of multiplicity X, so that aW(ax) ~ 0, t = 0,l, 2, ... , X - 1.
Thus for these roots,
dk^
a;«x(log™) (^^)aW(ax), 5 = X, X+1, ... , p-1.
Proceeding in this way we have finally for the roots equal to a^,,
dk^ /jfcrao
s = V, v+1, ... , l-l.
We have thus obtained asymptotic expressions for the solutions
belonging to the same group of roots of the indicial equation.
Since the roots aQ, ax, were arranged according to non-
descending order of their real parts, if Fs(cc), 75.^i(£c) denote
successive solutions belonging to the same group of roots, it
follows that
lim
Ixl-^oo
Vsjx)
= 0,
provided that x->co by a path inside the half-plane of convergence.
More generally when Fs(a;), Vi{x) are two solutions belonging to
the same group, then
lim
la;l'
VM
. F,»
-0,
provided that $ <Ct
If we consider all the solutions to which the indicial equation
gives rise, it follows that we can, in general, so number them, in
the order v^ix), ... , Vn{x), say, that
lim
i a; |->oo
'ys+i(a:)
= 0,
5 = 1, 2, ...
1.
14-3] RATIONAL COEFFICIENTS. OPERATIONAL METHODS 459
The linear independence of these solutions then follows from the
theorem of 12*16. We can therefore make the following state-
ments :
(I) The solutions corresponding to a congruent set of roots of the
indicial equation are linearly independent.
(II) When n solutions of a linear difference equation of order n
satisfy the conditions of 12*16, they form a fundamental set.
(III) If the indicial equation of a linear difference equation of
order n be of degree n, the corresponding set of solutions in general
forms a fundamental system of solutions. For, in general, they can
be so numbered as to satisfy the conditions of 12*16.
14'31. Series Solution Convergent in a Half Plane on
the Left. The solutions in series obtained by the use of the
operators tz and p, if they do not converge everywhere or nowhere,
converge in a half plane on the right, that is to say, in a half plane
which contains the point a; = + co . Any difference equation with
rational coeiffcients can also be prepared by means of the operators
7Ui and Pi for reduction to the second canonical form of Theorem V.
The series obtained from the equation so reduced will converge
everywhere or nowhere or in a half plane on the left, that is to say,
in a half plane which contains the point a; = - oo . The types of
solution obtainable in this way are
p^r(a;' + /c) f % ^2
r(x') { + +
. 1
^{x' + k-l){x' + k-2)(x'+k-Zy'"f
i^L^^^f^b, + bya/ + k) + b^(x' + k)(x' + k + l)
+ bs(x' + k)ix'+k+l){x' + k+2) + ...Y
where x' = x-r.
Example.
4:x{x+\)u{x-\-2)- 4:(p(?-\-x-{-l)u{x-\'l)^-x{x + V)u{x) = 0.
Put ^^(a;) = \L^v{x), x' ^ X and multiply by a?, then
4:[i^x pfv{x)-4: [i{x^ + X’{-l)piv(x)-\'X^{x+l)v{x) = 0.
460 THE LIHEAB DIFFERENCE EQUATION WITH [14-31
With --Ki + Pi for X, we have, from Theorem III,
a^ + x + l =K-pi)^-K-pi) + l = TCf-iri + l-(2TCi-2)pi + p2,
- x^x+ 1) = (tij- Pi)3 - (itj - pi)2
= TT® - Ttf - (Trf - 57Ci + 2) Pi + (S-rej - 4) pf - pf .
The coefl&cient of is 4^1+1. We therefore take [i =
and the equation reduces to
[7t| - TT^ + (tc^ + 3%) Pi] V (x) = 0.
For factorial series of the first kind, put
v{x} = aoPf+®iPi"^ + «2p?“^ + — •
The indicial equation is
(yfc+l)(jfc+4) = 0,
whence i = - 1 or - 4.
Series of factorials of the second kind can also be obtained by
putting
tj(a;) = 6opf+6ipf+^ + 6a pi+^+... .
14*4. The Complete Equation. Take the complete equation
(1) ZoW(a:) + ZiM(a:-l) + ...+Z„M(x-n) = X,
where Xq, , Xn, X are all functions of x.
To obtain solutions we can of course consider the corresponding
homogeneous equation and, by the use of Lagrange’s method, 12*7,
deduce a special solution of the complete equation from the general
solution of the homogeneous equation. Another method which is
more direct, when it is applicable, is the following :
Make the substitution u{x) = ]x^v{x) and reduce the equation
by means of the operators iz and p to the form
(2) [/o (tc) +/i {7r)p +/a (tc) p® + . . . +/^ (tt) p “] w (x) = p-* X,
the parameter p being at our choice.
If possible, we expand the right-hand side in one or other of the
forms
(3)
(4)
p,-®Z = Cfl p*-t-CiP»-l + C2p*^-2-l-... ,
= dg p*-(-dip*+^-fd2p*'^^+--- ,
MfM
14-4] RATIONAL COEFFICIENTS. OPERATIONAL METHODS 461
where the index h and the coefficients are of course known numbers.
When X has an expansion of the form (3), we assume that
v{x) = ■
Equating coefficients of like powers of p we obtain
fm(^) ®0 ~ ®0>
ai/m(^-l)+ao/m-i(^-l) = Cl.
a2/m(*-2) + ai/m-i(*-2) + ao/«-2(i5:-2) = Cj,
whence the coefl&cients can be determined successively.
If the resulting series converge we have a special solution of the
complete equation. To this we add the general solution of the
homogeneous equation. The given equation is then completely
solved.
When X has an expansion of the form (4) we put
v{x) = Bq + ,
and equate coefficients as before.
If [i-^X have expansions of the above t5rpes convergent in a
half-plane on the l6ft we use the operators .
14*5. Monomial Difference Equations. A difference
equation which after reduction by Theorem V assumes the form
(1) mu,^x
is said to be monomial.
It is evidently sufficient to assume that /(X) is a polynomial,
for the case in which /(X) is a rational function can evidently be
reduced to this.
Let /(tt) = a„ 71(71:- 1) ... (tt-w+I)
^a^^i7z{7z-l) ... (7r-w + 2)+,..-hao.
It follows at once, from Theorem II, that (1) can be exhibited in
the form
n
an(x-a) ,..{x-a-ni-l)
-1
,..{x-a-n + 2) A + = Z,
which is therefore the general type of monomial equation.
462 THE LINEAB DIEFERENCE EQUATION WITH [14-5
The monomial equation can be completely solved as follows.
Consider first the homogeneous equation
(3)
f{n) = 0.
This is satisfied by = p*, provided that Z; be a root of the
equation f(k) = 0.- If the roots of this equation be denoted by
ajL, Og, , a„ when these are aU distinct, we have for the comple-
mentary jpunction
XEfjP“>-f-...4-®„p“" = CTi
r(a;'-tl)
r(a;'-ai+l)
+ ... +
r(a:'+l)
r(a:'-a„ + iy
If the equation f{h) — 0 present a multiple root, we have
where a is a root of multiplicity s. We then consider the equation
f{^)u,=f{k)p^
which gives on partial differentiation .
. d*u.
/Ww-=.llC)
dk^ •
Since d^^'^f{k) j vanishes when
4 = a, V =: 0, 1, 2, ... ,5-1,
we see that, corresponding to k = oc, (2) has as solutions the values
of p*^, ^ p* , when k = x, that is to say, the solutions
r(x'+i) 0 r(K'-n) r(a:'-pi)
r(a;'-«-hl)’ 0ar(a:'-a+l)”"’ 0a»-i r(a:'-a4 1) ‘
We can thus find the complementary function in all cases.
To find a particular solution of (1) we have, symbolically,
Let us express 1 //(X) in partial fractions, so that
V _
A.
/(X) (X-a)
14-5] RATIONAL COEFFICIENTS. OPERATIONAL METHODS 463
Thea
by Theorem I.
To interpret the expression on the right, we have
p-“Z(rr) =
say. Then we can take
X
c c
and so on, so that (j)[x) = \fr (x), say.
Hence, finally,
a I / \ ^ 4- 1 ) f f ^
p“V-W = r(^/ra+i)^(^-^)-
The process simplifies if X can be expanded in the form
X = Cq p«+CiP<"“1+ —
We have then
w(a;) = = -7ftrp“+>> p°-^ + ...
/(t^) /(a) /(«-!)
by Theorem IV, provided that /(a) 0.
If, however, a be a zero of order 5 of f(Jc), we have
f(k) = {h-a.Y<l>{}c),
so that
1 1 1
• (7t-a)''^(7t)^ • {-K-ccy
= -8— 4 1
’ ^(a) 7c® ’
where the operation Tr*”* m now interpreted as explained above.
464
THE LINEAR DIEEEKENCE EQUATION WITH
[14-5
Exartifle.
S 2
2a;(2:- l)(a:-2) ls.u{x) + x{x-l) A u{x)->rx u{x)-u{x) = k?.
-1 -1 -1
Taking x' = x, this becomes
[2Tc(7r- 1)(ti-2) + 7t:(ti:-1) + k- l]M(a;)
= a;{a:- l)(a;- 2) + 3a:(a;- l) + a;,
(■nr-l)®(27t- 1)m(:c) = p + 3p®+p®.
Tor the complementary solution, we consider
(■n:-l)2(2-re-l)M(a;) = {k - 1)^ {2]c - 1) p’‘.
which is satisfied by
u{x) =
r(a:+l)
r(x-^+ij‘
Putting k=l,k = ^, we have
r(a:+l) r(a;4-l)
r(a:) ’ r(cK + |)'
Also
0 r(a:+l) _ r(a;+l) , .
d]cr(x-h+i)~T(x-k+i)^
Putting k=l, we get a:'P'(a;).
The complementary solution is therefore
cc (Wi+cTa’^^ (x)) + oTj r(a:+ 1) / r(a;+ |-).
For the particular solution, we have
u{x) = (^-l)-2(2TC-l)-i(p3 + 3p2+p)
= wP^+P^ + Ctc-I)"^?-
To interpret this we have
(tt:-1)”^P = pTt'^.l,
TT-il = -T"{a;|-l),
X
71-21= g
1
x-1
p7r-21 = a:giT'{tl-l)AL
If 5] RATIONAL COEFFICIENTS. OPERATIONAL METHODS 465
Hence the complete solution is
uix) = + {x)) + ©3 p|-— xj
X~\
+ ^^x[x-l){x-2) + x{x-l)+x ^
14*6. Binomial Equations. An equation whicli can be re-
duced to the form
[/o(^)+/m(7r) = 0
is called a binomial equation.
Putting u = aQ p* + p^-^ + . . . ,
we have % = aa = ... = = 0.
Thus we can assume that
-w = p^^ + 6p + pfe-27n_|.
and we obtain the indicial equation
fnci^ + m) = 0,
and the recurrence relation
bjra (h-{s-l) m) + 6,._J (/c - (5 - 1 ) m) = 0,
which gives
^ 1 )'/o - 1) - 2) m } . . . /o [k)
K 7m { h- {s-l)m }U {T- (5 - 2) m } . ..f2k) *
If one of the factors in the numerator vanish, the solution is given
by a finite, series.
Thus we can always obtain an explicit expression for the coefS-
cients of the series which satisfy a binomial equation.
The particular binomial equation
aiz+b
being at once reducible to an equation of the first order, can always
be solved in compact form.*
* We say that the solution is in compact form when expressed by a finite
number of operations of the form S-
466
THE LINEAR DIFFERENCE EQUATION WITH
[14-6
Again, tke binomial equation
[1 -a'^(j>{7z) = X,
where
(5i(Tc) = {(71-6) (tc- 6-1) ... (7r-6-?^^-l)}“^
can be written (see below) in the form
where s^, Sj, are the nth roots of unity.
If we put
t r S f] [' - Sl‘ ■=] ••; t - S <■] “ = W’
the given equation is equivalent to the n linear equations of the
first order :
[l-^jp]«2(») =
This is a particular application of the more general theorem that
the equation
[l + a^ ^(tt) p + a2^(^) ^(t^- 1) p^+*--
+ ^(tt) 1) ... <j>{7v- 71+1) u{x) = X
can be resolved into n linear equations of the type
[l-2r^{Ti:) p]w,(a;) = r = 1, 2, ... , n,
where Uq(x) = X, u„{x) = u{x), and ••• > ?n are the roots of
the equation
+ = 0.
We have in fact
[l-u.^(7r) p][l-6,^(,r)p]w
= [1~ (a-i-6) <l>{7z) p + a6 ^(tc) p ^(tc) p] w
= [l-(aH-6) ^(tc) p+-a6 «^(7r) (^(tt- 1) p^]^^,
and so on, whence the theorem follows by a simple induction.
4-7] RATIONAL COEFFICIENTS. OPERATIONAL METHODS 467
14*7. Transformation of Equations. The following problem
has been investigated by Boole. Given rational functions 9(X),
FCk) to determine, if possible, such that
P”}z(77).^= x(7c)J'{4;(ti:) p”}Z.
For tte special case F (X) = X, the above relation becomes
cp(7i:) p"x(’t).X:= x(^)4'(’^) P”-^>
or, by Theorem I,
9 (tt) X (tt - p« Z = X (n) 4^(tu)p^ X.
This will be satisfied if
9(X)x(X-n) == xW4'W^
which gives
^ ^(X) ^(X~^)+(X~2n)... ” *
Since Theorem I has only been established for rational functions,
we shall assume 9(X) and (|;(X) to be so related that x(^) is a
rational function.
With this value of x(X) we have then, denoting 1/xW by
ep(7i) p"Z = cp{TC) p”x(7r)X"MTc)-X’
= P"X~V^)-^
= X(7J:)+(7r) p"
Repeating the operation
[9 (tt;) p”]2 z = X (tt) 4^ {i^) p” x"’- X {'^) 4^ (^) p" x"’- ^
= X { 4' i'^) p" F x“^ (tt) X.
Continuing in this way we see that
[9(tu) Z = x('3t) {4'('^) p”}”* X”^(^)
The problem has thus been solved for F (X) = and hence for
any rational function jF(X), Thus we have proved the following :
Theorem VI. If 9(X), ^(X), jF(X) be rational functions, arid if
■ ■/-.s 9(X)9(X-to)9(X-2w) ... tt yW
~ 4;(X)4;(X-w)4;(X-2w)... ~ ■'■^”4/(X)’
then, 'provided that x(^) be rational,
F [9 (tt) p"] Z = X (77 )F [4^ (7t) p"] X"^ (77) X.
468 THE LIISIEAR DIEFEBENCE EQUATION WITH [14-7
A more general form of tHs theorem is obtained by assuming
X) to be a rational function of two variables (x, X.
In this case we have
F [pt, 9 (tc) p«] X (tt) = X (^) ^ ^ (^) P""] (^)
If we now replace p by tt, we have
Theorem VII. If 9(X), ij;(X), X) be rational functions,
and if
- y(x)y(x~^)9(x-2^)...
^ i]j(X) — 4^{X- 2n) ’
then, 'provided that x(X) rational,
F [tc, 9 (tc) p"] Z = x(7c) I’ [tt, (tt) p^] x“^ (tc) Z.
The reader will have no difiS.culty in proving the following :
Theorem VIII. If F{K), 9(X) be rational functions, then
jP[9(7c) p”]Z p^jF[9(7r + m) p”] p'^^^Z.
We now apply these considerations to the transformation of
equations.
The equation
(1) ^4 + 9 (tt) p’^ it = Z,
can be transformed into the equation
(2) v+^ (tt) p” ^ = Iln [if (^) / 9 (^)] ^
by the substitution
(A) '^ = Iin[9('^) I = XW^-
Tor making this substitution the equation becomes
X(7c) -yH- 9(7r) p” x(^) ^ = Z.
Operate with
^ (^) 9 (^) P"" X (■^) ^ = X""^ (^)
which by Theorem VI is equivalent to (2).
Similarly, by means of Theorem VII, we can shew that the
substitution
« = ni[?(7t)/'{'(Tc)]v
will reduce the equation
[/o(7^)+/l(^) P+/2(7i) 9(7^) <P(^- 1) P^]m = ^
14-7] BATIONAL COEFFICIENTS. OPERATIONAL METHODS 469
to the form
P +^2 (tc) i}j (tc - 1) p2] D = III [d; (7t) / 9 (7t)3 X.
Again, the substitution
(B)
will reduce the equation
U+ u = X
to the form
-y + cp (t: + m) t? = p-’” X.
This follows at once from Theorem VIII or can be proved inde-
pendently by means of Theorem I.
Boole has applied the foregoing considerations to the discovery
of conditions for compact solution, that is to say, solution by means
of a finite number of operations of summation, of certain equations of
the second order. We reproduce these discussions in full, as they
throw an interesting light on the structure of certain classes of
difference equations.
14'71. The Equation with Linear Coefficients. Let the
equation be
(1) (ax+b) u{x) + {cx+ e) u{x - 1) + {fx-h g) u{x - 2) = X.
If y 0, the linear change of variable fx-i-g = f(x'- 1) brings
the equation into the form
[ax' + b') u' [x') -h [ex' + e') u' [x' -1)4- J[x' -l)u' [x' -2) — X',
where b' = b-a[g+f)lf, e' = e-c[g-\-f) j f.
Suppressing the primes we may therefore consider the equation
(2) [ax-\-b)u[x)-\-[cx-\-e)u[x-\)-\-f[x-l)u[x-2) — X.
Putting u[x) = [L^v[x), pu[x) — xu[x-l), we obtain
(3) pi2^^^2^5^^^^3;)...j.j^|^^2ap4-c)7c4-(b-”U)piH'e]p'y(a;)
4- [ag? 4- cp. 4-/) p^ v[x) = x X.
If we determine pi so that
(4) agL^H-cg.-h/ = Oy
the equation assumes the binomial form
(5)
470 THE LINEAR DIFFERENCE EQUATION WITH [14*71
where
4 = 2 + ca-^ m = [(&-a)[J' + «][2ai^ + c]“^ n=ba-^,
V = [(at^) 7t{7i:+M)]-M^a!
and where we have assumed a ^ 0, 2a[i.+c =f=0.
We have here two cases of compact solution.
(I) Let m be an integer.
In this case the equation can be reduced by the substitution (A)
V {x) = Til [(’’■ ®
to the form
w{x) + A{Tt + n)-^pw{x) = ni[Ti:(n: + m)-i]7
or
{x+n)w{x)-{l-A)xw(x-l) = {n+n)Ui [tt {n + m)-i] V=W,
wHch is an equation of the first order whose complementary
solution is
OTi(l-A)®r(a:+l)/r(a:+w+l).
To this we must add a particular solution. From (6), we then
determine v{x) and finally u{x) = [xf u(a!) or [x*u(a;), where [Xi and
[12 are the roots of (4).
(II) Let m— w be an integer.
In this case we again use substitution (A) in the form
v{x) = ni[(u:+m){7i: + w)-Tw(a:),
which yields
vj{x)+Atz-'^^w{x) = ni[(7i: + n)(-n: + m)-i] V,
OT
w (a:) - (1 - A) w (a: - 1) = cc-i Tt 111 [(tc + ^
with the complementary solution (1 - 4)“.
(III) Eeturning to (3), let us, if possible, so choose [i that the
coefficient of p v(x} vanishes.
This is only possible if 2a(i+c = 0, (6-a)[i+e = 0, which
imposes tlie condition
2ae- (b-a)c = 0.
Supposing this to be satisfied, we obtain, with [i = - c / (2a),
u(a!)-^®7i:“’-(7t+w)~^p^ij(a;) = V,
14-71] RATIONAL COEFFICIENTS. OPERATIONAL METHODS 471
where
^2 — (<^2 — 4qjf) I y a'~^ 71""^ (tc + n)~^ x ul”“ X,
This equation has compact solutions if n be an odd integer
positive or negative, for with the substitution (A) in the form
v(x) = n2[(7c+n)-^(T[:- l)]^(ic)
(it+«)(» + «-2)... '
the operator is rational if n be odd.
The equation then reduces to
W{x)-¥7Z-^{7Z--l)-^fw{x)^ n2[(7r-l)"M7U+W)]F= IF.
Now
[1 - Ti^ 7r“^(7r- 1)“^ ^^']w{x) = [1 - {hiz~'^ P)(Atc"^ p)]^(-'^)
= [1 + A 7i:~^ p] [1 ~ ^ 71“^ p] w {x).
The further substitution
t{x) = [1-^71:“^ p]^(^)
gives
[l + hiz~^p]t{x)=W,
so that the solution is made to depend on two equations of the first
order.
This case is an illustration of the method explained in 14*6.
Example,
(X'{-2)u{x)-(x-\~2)u(x-l)-2{x-l)u{x-2) = 0.
Putting w(aj) = we have - [jl -- 2 = 0, whence
ji. = - 1 or 2.
Taking p. = - 1, m = I, A = 3, n == 2, we get
, 3(tc + 1) , . ^
Substitute v(x) = ni[(7r-fl)Tc“^]^«;(a;) = (71+ 1) then
'w(x) + 3(7v+2)-~^ pw(x) = 0,
(a: + 2) 'l^(^r) + 2^r^^;(^l?~ 1) = 0,
w(x) = (-2)«^r(a;+l)/r(^H-3) = (--2)^(x+2)-^(x+l)~\
v(x) === (7z-hl)w(x) = ^(-2)^-^(3x+4)(x+l)-^(x+2)-K
72 THE LINEAR DIFFERENCE EQUATION WITH [14*71
Therefore
u(x) =|2“^{3x+4)(a:+l)-Ha: + 2)-K
Taking (X = 2, m - 0, ^ = I, n = 2, we have
V (x) + -3 (u + 2)-^ p D (a;) = 0,
v{x) = (-2)“‘'(a; + 2)“^(a:+l)”^
u{x) = (-l)*{a: + 2)-Ha;+l)-^-
The general solution is therefore
, ^ ro,2»^(3!B + 4) «2(-l)‘'
“ (*) - (STl) (a!+ 2 j + (a: + 1 ) (a; + 2) •
14*73. Discussion of the Equation
(aa;^ + 6a;+c)M(a:) + {ea:+/)M{a;-l)+^M(a;-2) = 0.
Write u{x) = ii^v(x) / r{x+ 1), p = a; E"^ f^en the equation
becomes
y?{ax^ + bx + c)v{x)+{i(exi-f) pv{x)+g p^v{x) = 0,
whence, writing 7r+ p for x, we obtain
{ {aiz^ + &TC + c) + [X [ (2a [x + e) tc + (6 4* a) [x +/] p
+ ((ji2a+[xc + ^) 9"-}v{x) = 0.
(I) Choose [X so that
a[i?+e[i+g = 0.
The equation then assumes the form
[aiz‘^ + b'K-^c^-{A7z-\-B) p]'y(a?) = 0,
where
A\x = 2apL + e, B[x = (6 + a) pL+/.
This equation is formally satisfied by one factorial series of the
first kind and two of the second kind, all of which can easily be
obtained by our general methods.
If we put aB-^hh+c = a[h-~c£)[h-^), the equation can be
written
where
C = Aa-\ = BA-^.
14.73] RATIONAL COEFFICIENTS. OPERATIONAL METHODS 473
If either c-a or c- (3 be an integer, a compact solution exists.
If, for example, c - p an integer, the substitution
v{x)= ni[(7r + c)(7r-
leads to the equation
w{x) = C{7Z'-ay'^ pw{x),
which is of the first order.
(II) When the coefficients are related by the equation
2af + (a - 6) c = 0,
we can choose [ji so that
2<X(jl + e = 0, (6 - a) p +/ = 0.
Thus putting pt = - e / {2a), = (e^ - iag) / the equation
becomes
V (x) ~ (tc - a)""^ (tc — p)“^ V {x) = 0.
■ If (3- a be an odd integer (positive or negative), the substitution
(A) in the form
v{x) = n2 [(tu- (3)"^(7r-a- l)]?^(a;)
leads to the equation
t4;(a;)~7r(Tc-a)~^(7u~a-l)“^ p^w{x) = 0,
which can be resolved into two equations of the first order as
in 14-6.
Example. {x^-\-x-2)u{x)~{-bxu{x-l)-\-iu{x) = 0.
Putting u[x) = [i^v{x) /r{x + l), the equation becomes
[(j,^(7i^ + 7r- 2) + p.(2fx + 5) 7rp + ((j[,2 4-5p. + 4) = 0.
If [jL = - -.5, this gives
Put
v{x) ~ ri2[(7r+I)(7i:-l)“^]^(^(a;) = (tc + 1) 't«;(a;).
[I - -2^- {tc + 2)-^ (tt 4- r)”^ p^] (^) = 0,
Then
474 THE LINEAR DIFFERENCE EQUATION WITH [14.73
which, gives the pair of equations
[1 — (tc 4 2)""^ p] t {x) = 0,
[l + -|(7c42)-"ip]'w;(x) == t{x).
The first of these gives
A5/ T{x+Z) V5/ (a:+l)(a; + 2)'
The second then becomes
(x+2)w(a:)-fxM(a;-l) = 5^+1)’
which can be solved by summation. We then have
u{x) = (~-|-)®(:t41)
t4‘76. Discussion of the Equation
2
{ao!^-\-bx-\'C) ^u{x)i^{ex+f) /\u{x)+g u{x) = 0.
This equation can be written in the form
2
a(x-oi){x--i^) /!s^u{x) + e{x-’Y) /S,u{x)+g u{x) = 0.
Here it is convenient to use the operators tti and p^.
Taking x' = x- a - 1 and multiplying by x',' we obtain
a(a;'4a- p4l)7Ci(';ii-l)w(a:;)
+ e(x' + (x.-y + l)7z-^u{x)'^x' gu{x) = 0.
Write - 7Ui4 Pi for x', we then obtain
[a(--TCi4a- p + l4pi) 71:1(711- l)4c(-7ti -ha - y + 1 4 pi) 7ti
+gr(-7ri4pi)]w(x) = 0,
which, using Theorem I, becomes
7Ci [a (tci “ a 4 P — l)(7ri- l)4 6(7ri — a4 Y~l)4^]'u(ir)
~[a(7ri-l)(7ri-2)4e(7ui-l)45^]piw(a;) = 0,
which is a binomial equation formally satisfied by three series in
ascending powers of pi and by two series in descending powers
of pi, all of which can be found by the usual method.
U-8] RATIONAL COEFFICIENTS. OPERATIONAL METHODS 475
14-8. Bronwin’s Method. Certain forms of linear equation
can be solved by performing A upon them one or more times.
Consider the equation
2
(a + 6cc) + = 0*
n
Operate with. A* Then by the analogue of Leibniz’ theorem, 2*51,
we obtain
w.H-2 71+1
[a+hix+n)] A u+nh A «
71+1 Tit 71
+ [c + <i(cc+n)] A + A'W~i-<2 A'^ = 0.
If we take n = - ej d, supposing that to be a positive integer,
71+1
we have a linear equation of the first order for A
2
Example, ccA'^ + (2^“2)A^-'^ = 0.
Performing As we have
3 2
(^^+1) A^+^A'^ = 0,
whence
2 w
A“=
c
Substituting in the given equation, we have
C
14-9. Linear Partial Difference Equations. The prin-
ciples of solution enunciated in 13*8 are applicable to partial
equations of the following forms, namely,
F{x, Ax7 Av) F{y, Aa;, Av) ^
F{x, y, Aoj)^ = Av) = 0.
In each of these equations one of the independent variables
or one of the partial operators is absent. If y or Ay be absent,
we treat y as constant and the equation as an ordinary equation
in X and thereafter interpret the solution.
476
THE LINEAR EIPFERENCE EQUATION WITH
[U-9
Exmnple, , y^l = 0.
This equation can be expressed in the form
= 0.
Replacing Ey ^ by a, - we have
u-ax JE.X 0,
which is equivalent to
= 0,
where y, We thus obtain
v^ = a^T(x+l)<f>{y),
and therefore
= Ey~ T (^4- 1) (y) = V {x 1) (j> {y -- x) ,
where i is an arbitrary function.
14*91. Laplace’s Method. The method of 13-83 is appli-
cable to equations of the form
AqU{x, y) + AT^u.{x-l, jr~l)+-^2w(ir-2, t/~2)4-...= Vix, y),
in which the difference of the arguments in u{x-s, y-s) is in-
variant for 5 = 0, 1, 2, ... ,
Putting x-y = 7c, we obtain
BqV{x)-{-B^v{x-1)-^,..= V{x, x-lc),
in which the coefficients are functions of x and of the parameter Ic.
Thus in the equation
^£Cj y ^ J , 3/— 1 ~ Oj
we have
whence %:=cV{x+l),
Replacing c by an arbitrary function of Ic, we get
as before.
'^x, y = r 1) </> (as - 2/)
BATIONAL COEFiriCIENTS, OPERATIONAL METHODS 477
[BX. XIV
EXAMPLES XIV
Investigate the solution of the following difference equations :
1. {x+‘l)u{x + 2)-xu{x+\)-u{x) = 0.
2. (a; - a) - (2a; - a - 1 ) M-„_i + ( 1 - (a: - 1 ) = 0.
3. (a: + 3) m*+2 - (a: + 3) - 2a: = 0.
4. a:(a:+l) A“-2a:AM + 2w = a:(a:+l)(a; + 2).
6. a:(a:+l)M*+2-2(a: + 2)a;«^+i+(a:+l)(a:-2)Mi„ = 0.
6. _ 1) (a; + 1) _ {x^ + x-f 1) (a:® + a: - 1) w«+i+ x^ {x + 2)
= 0.
2
7. a;(a:+l) Aw + ajAw-’^®^ = 0-
8. (a: + 2) (2a: + 1 ) «^+2 - 4 (a: + 1)® + x(2x+ 3) % = 0.
9. M(a: + 2)-a:2M(a:+l)-c(c-x2)M(a:) = 0.
10. ^a:4'2 X^ 0.
11. (a:24.(ja;)^u_(2a:+fl-l)A«+2M = bx+ca^.
12. % = a:(Mj,_i+Wx_2)-
13. u^+i = a:(Mx+Wx_i).
14.
(a:+l)^(a:+2)
16. Shew that the equation
can be reduced to a linear equation of the second order by the
substitution
and shew that the two periodics which appear in the value of v^.
effectively produce only one periodic in the value of m* •
16. Mx+2 - 2 (a: - 1) Wx+1 + (a: - 1) (a: - 2) Mx = r (a: + 1 ).
2
17. a:(a:+l)Aw + ^:(l-a:)AM + *« = 0•
18. Ux+i,»+i-(a-a:-22/-2)«*. ,+i + (a:+y)«*.i/ = 0-
CHAPTER XV
THE UMAR DIFFERENCE EQUATION WITH RATIONAL
COEFFICIENTS. LAPLACE’S TRANSFORMATION
In tMs chapter we discuss the application of Laplace’s transform-
ation to the linear equation and the solution by means of contour
integrals.
15-0. Laplace’s Transformation. Another method of
solving difference equations with rational coefficients is founded
upon the substitution
(1) u{x)=\t^'^v{t)dt,
J I
where I is a line of integration suitably determined and where the
function v{t) is found from a certain differential equation. As
all the essential points of the method are illustrated by the equation
of the second order we shall consider the equation
(2) p^{x)u{x+2)+Pj^{x)u{x+1)+j)^{x)u{x) = 0,
where pi{x), fi{x), Pq{x) are polynomials.
An equation of this type wiU be called “ normal ” if the following
conditions be satisfied.
(i) The extreme coefficients have the same degree p while
that of the remaining coefficients does not exceed p.
(ii) The differential equation satisfied by v{t) is of Fuchsian type,
that is to say, all the singular points of the differential equation are
regular.*
* The somewhat unfortunate term “ regular ” is here used in the sense in
which the term is applied in the theory of linear difierential equations. See
e.g. E. Goursat, Gours d’ Analyse, t. ii (2nd edition), chap. xx. The term
regular must not be confused with holomorphic.
478
15*u] COEFFICIENTS. LAFLACE’S TRANSFORMATION 479
We shall suppose equation (2) to be normal.
We then write the coefficients in the following form :
7^2 (^) ~ Ap(x~\-2) 4- 3) ... + 1) + ...
-r^2{^‘h^) (^ + 3) 4-.^l(iC4*2) + i4Q,
Pi (a?) = B^{x-i-l){x + 2) ... (a;+p) + ...
-\-B^{x+l){x+2)-{-B^ {x+1)'+Bq,
Pq(x) = C^x{x+l){x + 2) ... (a;+p-~l) + ...
"h ^2 ^ "t 1)4- C^i X-^-Cq,
where Aj,=^0i
Putting
4>‘p{f) = j4j, + Cj,,
(3) </>,(0 = .4,i2 4.j5.i + C,, i = 0, 1, 2,...,p™l,
the equation
Ut) = 0
is called the characteristic equation. By our hypothesis the roots of
the characteristic equation are both different from zero. We shall
denote these roots by a^, a^.
With the value (1) for u{x), we have by successive partial in-
tegrations
a;(cc4-l) ... (cc4-s-~ 1) {t) dt
4-[(a;4-5-l) ... (cc-l- 1) ^;(i) - (x + s- 1) ... {x + 2)t^+^v'{t) + ...
Substituting for u (x) in (2)^ we obtain for the left-hand member
the expression
s^Q
where
(4) I{x,t) =
•f ( - {t) (t)].
It follows that (1) provides a solution of the difference equation
(2) if V (t) be a solution of the differential equation
(5) ^ = 0,
480 LINEAE DIFEEEENCE EQUATION WITH RATIONAL [is-o
and if the line of integration I be chosen so that I (x, t) has the
same value at each extremity of the line, when the line is open. If
the line be closed, I {x, t) must return to the same value after t has
described the line.
The singular points of the differential equation are i = 0, { = oo
and the zeros of that is to say, the points t - a-^, t — a^.
To find the solutions of (5) in the neighbourhood of the ori»in
we substitute
v{t) — Coi’" + Cii”*+^ + C2U"+2+ ... ,
and equate to zero the coefficient of This gives, as indicial
equation for m,
Co[Cj,m(m-l) ... + ... {m-p + 2)+...
which, by the definition of {x), is equivalent to
Poi-m) = 0.
Thus if, as in 12-0, we denote the zeros of p^(x) by oci , Kg , .. . , a^,
we have as values of m, -a^, -Oj, ... , -a,. We shall suppose
these values arranged so that
J?(ai) < R{a.^) < J?(a3)< ... :^ i?(a,,).
The differential equation has then p solutions in the neighbour-
hood of the origin of the form
(6) v,{t) = [ foil) + flit) logt+...+f,(t){logt)-],
s=l, 2,...,p, r<^p-l,
where the functions /o(i),/i(<), ... ,f^{t) are holomorphic at i = 0.
If no two of the numbers % , aj be congruent, no logarithmic
terms occur. In the extreme case where all the are congruent,
r = p - 1, when s = p.
Again, the j^roduct vanishes when t = 0, provided that
~ a^j) ^ 0. This condition is satisfied for every s, provided
that R{x-ol^)> 0.
Now, the functions v^(t), (5 = 1, 2, , p) form a fundamental
system of integrals of (5). It follows that any integral v{l) is of
the form
(^) = '^1 (0 + ^2 ^2 (^) + . . . + 6^ Vp (t),
where 63, ... , are constants.
15-0] COEFFICIENTS. LAPLACE’S TRANSFORMATION 481
Tiius t^v{t) vanishes when i = 0, provided that i? (x - > 0.
It follows, from (4), that I (x, t) vanishes when t = 0, provided
that R{x-a^) > 0.
To examine the point <5 = cx) , we put
v{t) = ,,, .
This yields the indicial equation
Aj,m{m-l) ... {m-p-i-l)- ... (m- jt? + 2)+ ...
+ ( ~ = 0,
which is equivalent to
P2(“-w^-2) = 0.
If we denote the zeros of ^2(ic~2) by Yi, yg, , Yj,, arranged
so that J?(yi) ^ ^(Y2) ^ ^ •2f2(Y2))j we have for m the values
“ TiJ “■ Y2j ••• j ”■ Tj) ^ fundamental set of solutions of the form
F,W = t-y^[s,(t)+g,(t) log t-h,..+gr{t) (log 01,
5=1, 2,...,p, r^p-l,
where ^0(0? ••• > 9r{^) are holomorphic at infinity.
It is clear then that vanishes at if = 00 , provided that
J?(aj + 2-Y2))<0, 5= 1, 2,...,;p,
and, if this be so, we conclude in the same way as before that Z (a;, 0
vanishes at 1= 00 ,
It remains to discuss the singular points t =: a^, t ~ a2.
Two cases can arise ;
(i) ag; (ii) % = ag.
In the second case the differential equation is of Fuchsian type if,
and only if, be a zero of We shall suppose this to be the
case, so that the difference equation is normal in accordance with
the definition.
In the differential equation, substitute
V (0 = Co (jf - %) + Cl (^ - aO '^+1 + Ca (i - aj ) + . . . .
In case (i), the lowest power of {t ~ a^) is {t - Equating
to zero the coefficient of this, we have the indicial equation
(8) (ai)m(m-l) ... (m-j3+l)Co
... {m-p + 2) Co = 0,
482 LINEAR DIFFERENCE EQUATION WITH RATIONAL [15-0
which gives for m tlie ^ - 1 integral values 0, 1, 2, . . . , ^ 2 and one
other value which we denote by Pj. The solutions corresponding to
the integral values of m are holomorphic at i With these
solutions we are not concerned. On the other hand, is not, in
general, an integer and the corresponding solution is
(9)
where /i(^) is holomorphic at i
Similarly, at t = we have p-l holomorphic solutions and
another solution
(10) 'V*{t) = {t-a^Y^f^(t),
where Pg is i^^t, in general, an integer and /g {t) is holomorphic at
t ^ Qf^.
In case (ii), the indicial equation for m is
+Com(m-l)...(m-2)+3)^3,_2(ai) = 0,
which gives for m the y-2 integral values 0, 1, , y- 3 and two
other values not, in general, integral which we denote by and Pg
with J2(Pi) ^
The corresponding solutions are of the forms
(11) vl{t) = {t-a^)^m,
•v* {t) = (t- [/s (0 +fz (i) log (« -«!)]>
where fz{i) are holomorphic at t = The logarithmic
term will only occur when P2 congruent. f
15*1, The Canonical Systems of Solutions. The integrals
^2(^) differential equation 15*0 (5) are many- valued
functions.
t For the method Frobenius applied to the case of congruent indices, see
Fors;^h, Theory of Differential Equations^ vol. iv (1902), pp. 243-258. The
solutions can be written
0fc-l
where h=l, = k = 2, m = ^z.
15.1] COEFFICIENTS. LAPLACE’S TRANSFORMATION 483
Two ways in which v* (^), v* (t) may be made one- valued
are shewn in Figs. 15 and 16. In Fig. 16 we have cut the t plane
Fig. 15. Pio. 16.
from 0 through to infinity and from eig infinity along the line
Oug- In ®nt plane both the above functions are single- valued.
Fig. 16 is explained in the same way. In the figures and Zg are
loops from the origin round % and Ug respectively.
If R(x- ocp) > 0, we have seen that the function I (cc, t) given by
15*0 (4) vanishes at the origin, so that
(1) Ui{x) = ^^^t=‘-^v*(t)dt,
(2) W2 (a:) = 2^ v* (f) dt
are solutions of the difference equation 15*0(2). These solutions
have been called by Norlund the first .canonical system. That
these solutions are linearly independent and therefore form a funda-
mental set will be proved later.
If ag = r2e'^\ 0 0i < 62 < 27r, we shall suppose that
as t describes in the positive sense starting from 0,
increases from 61-71 to 61 + tt:, and that as t describes Zgj arg(^-a2)
increases from 62 -tc to 02+tc, while argi = 0i or Og along the
straight parts.
If namely, OaiUgCo , is necessary, and only
dne loop circuit is needed for both solutions. If 61 = 62 j
both % and he on the cut, and we must deform the loop Zg so
484 LINEAR DIFFERENCE EQUATION WITH RATIONAL [I5 I
that the point % is not enclosed by l^. In every case each loop
must be drawn so as to enclose only one point which represents
a root of the characteristic equation.
Consistent with this, the loops may be of any shape, the most
convenient shape for say, generally being two straight lines
ultimately coincident with 0% and a vanishing circle round a^.
We note also that, if JS(Pi)>~l, the integral taken round the
circular part tends to zero when the radius of the circle tends to
zero. In any case we shall suppose that neither nor pg is an
integer, for in this case the integrals taken round the loops vanish.
It may be noted here that integrals taken round a double loop
contour (such as used for the Beta function in 9-89) joining any two
of 0, ag will furnish a solution of the difference equation.f
The second canonical system of solutions is furnished by cutting
the plane as already described and taking infinite loops and
round and as illustrated in Fig. 17 for the point a^.
This gives
(4) ^2 = 2^ ^2 (*')
where along L-^ we take SiXg{t-aj) to vary from along the upper
side of the loop to 2n+ 6i along the lower side, arg t being 6;^ along
the straight parts. Here again the shape of the loops is immaterial,
t See E. W. Barnes, Messenger of Mathematics, 34 (1905).
485
15.1] COEFFICIENTS. LAPLACE’S TRANSFORMATION
provided eacli encloses only one point representing a root of the
characteristic equation.
The second canonical system of solutions also forms a fundamental
set.
15*2. Factorial Series for the Canonical Solutions. We
consider for simplicity the case in which the roots of the charac-
teristic equation are incongruent.
We have by 15-0 (9),
where J{t) is holomorphic at t =
The only singular points of -y* {t) inside and upon are t =
f = 0. Thus f[t) is holomoiqihic inside and upon \ except at i = 0.
Make the change of variable
t — co>l.
Then
where C is a loop from the origin round 2=1, Fig. 18.
Now the circle | 2 - 1 | = 1 in the 2: plane transforms into a loop
in the t plane, round t = enclosed by two rays OA, OB inclined
at angle tt/o.
For, if
2 = r t = T % = ri
we have
486 LINEAR DIEFEBENCE EQUATION WITH RATIONAL [15-2
SO that.
— 0 -j- 6).
When P describes the circle, 6 varies from - 1 to + 1 and r varies
from 0 through 2 to 0, so that 4^-63. vanes from “ + ^>
T varies from 0 through ri2i/“ to 0, so that T describes a loop of
the kind stated. 1
By CO large enough, that is, by making the angle AOB
small enough, we can ensure that 0, a, are the
points of uf(«) on or inside this loop, and consequently thay{f)
is holomorphic in and on the loop except at t-0. It
that f{ajfl“) is holomorphic inside and on the circle 1 z - 1 j 1,
except at z = 0.
We can therefore find an expansion
2Tcico^ ^ ^
which is convergent inside the circle [ z - 1 1 = 1, so that
% (^) = ( af (2 - 1)^' s
J (J V — 0
Since the loop C is interior to the circle, we can integrate term
by term. Since by hypothesis arg{z- 1)= - ir at the beginning of
the loop,
u, {X) = e-«^> a? E a f (1 - 2)^’+''
_ g-TOft (1 _ eSTift) 2 C„ B , Pi + V + 1) ,
v=0
in terms of tlie Beta function from 9*88.
Since
B(a:/oi,Pi+v+l) = r(a;/co)r(pi+v+l)/r(a;/oi + Pi + v+l),
we can write
Ui(x) = af r(£i: / co) !:2(a:. Pi) / r(x/co+ Pi + 1),
ig.2] COEFFICIENTS. LAPLACE’S TRANSFORMATION 487
where (l{x, pj) is a factorial series of the form
nt r.\ n I 7) (Pi+l)-(^i+^)
f2(a;, Pj) -Uo+ (a;4.a)pj + co)...{!E+coPi+v«)’
where I>o=f=0 and where the series is convergent for R{x~ol^) > 0.
In the same way by means of the change of variable
t = ai(l-2)-i'‘“,
we can shew for the second canonical system that
U^ix) = o® 2 Pi, Pi+ v+l).
V =0
15-3. Asymptotic Properties. If x— >oo in the region of
convergence, since
rjxjoy)
r(x/co + PiH-l)
we have
Mi(x) ~ af(a:/co)-^> ' Z>o
Similarly,
Mg {x) ~ (x / co) “
Hence, if
|ail> l«2l>
ito = 0,
z-^00 % W
so that, by the theorem of 12-16, the first system of canonical
solutions forms a fundamental system.
The same can be proved for the second system of canonical solu-
tions.
As we have thus found two fundamental systems, it follows that
the members of one must be linearly related to those of the other.
General methods of finding the periodic coefficients of these relations
have been developed by Norlund. They are too long to introduce
here but they will be illustrated by an example later.
D{x)
488 LINEAR DIFFERENCE EQUATION WITH RATIONAL [15*31
16’ 31. Casorati's Determinant. By Heymann’s Theorem,
12*12, the determinant
Ui{x)
u^{x+\) u^ix + l)
satisfies the difference equation
■0(^+1) _ f^{x) _ C^{x~OLy) ... {X-OL^)
(l^ Qj^ '
*4„(a:-Yi + 2)...(x-
h, say.
t7+2)’
B{x)
Hence we have
\ z,^r(a;-ai) Ffx-ao) ... rfx-aJ , ,
Dix) = ^
r(ir-Yi + 2) ... r(x-Y^ + 2) ^
where w (a?) is a periodic whose value will now be determined.
By the asymptotic formulae for u-^{x), u^{x), we have
= af Z)o(l + 7]i (a;) ),
u^{x) Dl^{l-^yi^{x)),
where 7]i(cc), r^^ix) —>0 when x ~> co in the region of convergence.
Now
+ + (a?+l))
= -Do (1 + v]3 {x) ) ,
where y)^ (x) -> 0 when . a? ->■ oo . Thus
D (a?) = Dq {a^ - %) (1 + 7] (x) ) ,
where t] (x) -> 0 when a; — > oo , so that
D (x) — b^ a;~^i-^2-2 j) j)'^ ^ \ ^
But
r{x-y, + 2)
so that, from the value of B{x) iu terms of the Gamma functions,
we have
B{x) ~
where
p
* = S (y.'i-as)-
<s — 1
Comparing the two asymptotic values, we have
mix) ^ (a^ - a, ) .
15*31] COEFFICIENTS. LAPLACEVS TEANSFORMATION 489
We shall now show that the index of x is zero. From the ex-
pressions for we have
Thus
Again, from 15*0 (8),
Y» p-i + i ■p{‘P-'i)A„) I A
1
30 that
_ 1 _ A
(1 A '
\/in * p
Q,-p + l = %4“v
Pl+ Po- 2p + 2 = H- .
If we express (/>p^i(x) / in partial fractions, wo get
^33-1 (^) _ 4^p-i fa) 9^3) -1 (^2)
i>p {^) ^3? “ <^i) (%) — ^2) (^2)
Putting cr = 0, we ol)tain
Ap^i
and therefore
Thus
2p~-2-^,-^,-/c=0,
z!j(x) ^ BoDoia^-aj^).
It follows that the periodic w (x) is a constant whose value is
Z)o-DoK-%)*
Thus we have the value of Casorati’s determinant, namely,
i)(a;) =
V(x-oLi) ... r(jr~aj5)
f^^r:^4:2^..r(a;~-y, + 2)
6® Z)0 Z)o (<3^2 ~ ^1) •
The above result, which is due to Norlund, has been obtained on
the assumption but even if % = Norlund f has shewn
that m[x) still reduces to a constant.
t N. E. Norlund, 'Equations linkLires aux differences finies (1929), chap. iii.
490 LmEAR DIFFERElSrCE EQUATION WITH RATIONAL [15-4
15*4. Partial Fraction Series. The canonical solutions, as
is evident from their developments in factorial series, are anal3rbic
except at poles in the region of convergence. By means of the
difference equation itself these solutions can be prolonged over the
whole plane. It thus appears that these solutions are meromorphic
functions and must therefore, in accordance with a theorem of
Mittag-Leffler, have a representation by a series of partial fractions.
To obtain the development we make use of the solutions of
15*0 (5) in the neighbourhood of the origin.
On the line Oa^, take a point a which is nearer to 0 than ag is.
Denote by G the path of integration OaaO and by C' the loop
from a round % and back to a. Fig. 20.
Then
2ni 'Wi (x) = 1 vf (t) dt + f {t) dt,
Jc JC'
The second term on the right is an integral function which we
can denote by 2iiiE{x),
In the first integral, using 15*0 (7), we have, on the path Oa,
while on the return path aO we have
where 65, (s = 1, 2, ... , p) denote two sets of constants corre-
sponding to the two determinations of vf (^) on opposite sides of the
cut in the t plane.
154]
COEFFICIENTS. LAPLACE’S TRANSFORMATION
491
Thus
f u? (t) dt='£t {b, - e,) f V, (t) it.
J c « = 1 *^0
Now, by 15-0 (6),
(i ) z= {-“s [fo(t) + A(t)logt+... + A W (log i)'] ,
where /o(0>/i(0: ••• > holomorphic in the neighbourhood of 0,
and consequently along the path Oa, since
|aj<lai| and lal<la2l.
lyi-^ln'Tig the change of variable t = az and expanding the holo-
morphic functions in powers of z, we have
f t^-^vt{t)dt - a" i; 2 S r
27rtJc stroll K=o Jo
where the are numerical coefficients.
Now
Ttus, with 1)*" v! we have
^ 'P r jg
u,{x) = E(x) + a^ g S g =
which is the required development of u^(x) in a series of partial
fractions. The series converges over the whole plane except at the
points a,,a,-l,a,-2,...,(s = l,2,....:p) which are poles. If
no two of the be congruent no logarithmic terms can appear
in the functions v,{t) and we have in this case the simpler
development
QO ^ B
Uj^{x) = E(x)+a^ ^ (jP+s-aj
15*5. Laplace’s Difference Equation. This name is given
to the equation whose coefficients are Unear functions of a;. The
general form of the homogeneous equation is
[A^{x+n) + A^'\u{x+n)+[Bi{x+n-l)-hB2]uix+n-l)
+ ... + [KiX+K^u{x) = (i.
492 LINEAR DIFFERENCE EQUATION WITH RATIONAL [15*5
If = J5i = ... = jEli = 0, we tave an equation with, constant
coefficients. Some cases of the application of Laplace^s trans-
formation have already been discussed in 13*3. For simplicity we
shall again consider the equation of the second order
AqI w (a?+2) + 1) + Bq] (a? h- 1)
+ [Oi X + Cq\ u (x) == 0.
If Ai 0, 0, this equation is of the normal type.
Making the substitution
we have for v{t) the differential equation
ix {f)~ (Aq + Bq 1 4" Oq) V {t) = 0.
When the roots a^, of the characteristic equation
are unequal, we have, using partial fractions,
^'(0 - I Pi I P2
v{t) t t-
{t) = (t - (t -
where
C'loc + C'o =
Again, from 15-0 (4),
I{x, t) =
The canonical solutions are
^ ^ 1 “2 (») = 2^^ V*{t)dt;
= U,{x) =
The expressions for u^{x), u^ix) are valid in the half plane
R{x-a.)>0; those for V^lx), U^ix) in the half plane
i2(a5-a+pi + P2) <0.
15-61] COEFFICIENTS. LAPLACE!S TRANSFORMATION 493
15-51. ReducibleCases. Since «* («) is multiplied by
passing round t=a-y, we have
If Pi be a positive integer or zero, we see that /(a;, t) vanishes
when i = «!, so that we can replace Wi(a:), which vanishes, by
0
In this case if we make the change of variable t~ a^z and for
brevity write ^ = a;-~a, we have
“2(2:) = +
where I denotes a loop from the origin round z — I .
Expanding the last term of the integrand by the binomial theorem
in powers of z-1, we have
«2(^) = ^ t (1 ' (s)
Thus, from 9*88, we have
U2(x) =
^=0 ^
'Pi r(^)
Vs/ r(^4- P2+S+ 1) “ P2~ ®)
= a f' fPi'i r(g)(^+p8+s+i)-(^+p2+p,)
r(?-l-pi+p2+l)r(-P2-«)
Hence, when jSj is a positive integer, we have (omitting a con-
stant factor),
Mo (x) = af p (x),
’ 2r(a:-oc+pi+P2+l) ^
where
P{x) = {x-h^{x-k^ ... {x-k^^)
is a polynomial of degree [3i.
It follows that
u^ix) = af
r(a;-a)r(a;-Ai+l)...r(a?-^^,+l)
rCir-a+Pi+Pa+l)^!^?-^^!) ... r(a;~ifepj '
494 LINEAB DIFFERENCE EQUATION WITH RATIONAL
Thus u^{x) satisfies an equation of the form
^^(a^;+l) = r{x) u{x)^
[15-61
where
t{x) =
(a; -- a + Pi + P2 + 1 ) (a; - ifci) . . . (a; ~ J ’
and the given diifference equation is reducible (see 12-24).
Again, if Pi= a negative integer, we have
u^{x) :
_1_ f
2TzijL
dt.
Hi —
The residue of the integrand at the only pole <5 = ^1 is the coejffi-
cient of in {y + aj)^^^{y+kY^, where This is
so that
Ut^{x) a\P [x),
where P (a;) is a polynomial and the equation is again reducible.
Thus Laplace’s equation is reducible when either or pg is an
integer.
15*52. Hypergeometric Solutions. We suppose that
neither nor is an integer and that | % | < | | . We have
then a solution of the form
tq(a5) = I
where ^ = a: -a. Putting i = Oj z, this becomes
where { is a loop from the origin round 2; = 1.
Since the integrand is multiplied by on passing round 2: = 1,
we have
Jo
15-52] COEFFICIENTS. LAPLACE’S TRANSFORMATION 495
from 9*86. This is valid, provided that
These restrictions can be removed if we replace I by a double
loop of the form used in 9*89. By various changes of variable we
can get altogether 16 solutions of the above type. Eight more can
be obtained by taking a loop from round and a loop from
round a^.
On account of the existence of these solutions, Laplace’s difference
equation of the second order is also known as the hypergeometric
difference equation.
15*53. Partial Fraction Series. Taking ja] less than
I aj I and ] Ug |, we have, as in 15*4,
Uj^{x) =1 ^ (a^ - (a^ - dt
= (i-e2«^i) {a2-t)^dt+E{x),
where E{x) denotes an integral function.
Now, by the binomial theorem,
We can put | a j equal to the smaller of [ % |, | ag |.
If 1 ai I < I ag 1, > - L tlie integral round vanishes, and
we have
The partial fraction series are valid in the whole plane and put in
evidence the poles at cc = = 0, 1, 2, —
496 LINEAR DIFFERENCE EQUATION WITH RATIONAL [15-o4
15-54. The Relations between the Canonical Systems.
Let flx = ®2 = ^2 0 < Oj < 02 < 2tz.
Consider the loops l-l shewn in Fig. 21, where the straight
parts coincide with Oo^ and the radii of the circles tend to zero.
We first suppose that
i?(^)>0, 22{W>-1,
and put
XW =
On AB let arg t = G^, arg(i ~ = 0i- tc, then
271:^ u-^{x) = (1 - | \{t) dt,
Jo
since the integral round the circle tends to zero with the radius.
On EF arg {t - %) = 63^ + tu, arg t = G^,
while on GH arg i = 6^ + 271, arg {t - a^) = G^ + tu.
Hence
[ X (0 = f X (0 X (0
J J • J 0
Thus, comparing these results, we have
(1) 2niMx) = (f)*.
This integral has a meaning even when R{^) < 0, so that we have
obtained the analytic continuation of Uj^{x) over the whole a; plane,
except of course at the singular points. We can therefore now
suppose that (3^+ pg) < 0. Without crossing any of the cuts
in the t plane (see Fig. 15), we can enlarge the loop into a large
15-54] COEFFICIENTS. LAPLACE’S TRANSFORMATION 497
indented circle whose centre is the origin as shewn in Fig. 22, and
when the radius oo since <0, the integral along
the circular arcs will be zero.
Let M, Q be supposed to coincide with and Nj P to be at in-
finity along 0%. We then have on MN, arg(^-ai) = sbxgt =
and on PQ, arg {t - a{) = + 2tc, arg = B^ + 27r. Thus, if X denote
the limit of the contour PQa^ MN when the radius oo , we have
_ p 2tra ^ (i) + j X (0
X J 00 J cti
Dealing with the canonical solution (x) in the same way as with
Ui{x), we obtain
2Tci C7i (;;c) = f X (0 X (0
Thus we have
JL
2ni
v^{t) dt
I _ g2Tri(f+j3i)
U,{x).
Also the loop round ag is equivalent to -Lg.
We have now the complete value of the integral round 1^' and,
substituting in (1), we obtain
Ui{x) =
1
UA^)-u,{x)]
This is an identity between analytic functions and we can there-
fore remove the restrictions originally placed on Pa* The
498 LINEAR DIFFERENCE EQUATION WITH RATIONAL [15-54
expression of U2{x) in terms of Ui{x), is obtained from
the above by interchanging the suffixes 1 and 2 and then writing
g2irif U^(x) for TJi{x), the factor being introduced on account of
the additional circuit round ^ = 0.
This investigation illustrates a general method of finding rela-
tions between the solutions of the two canonical systems.
If and r -> oo , then -> 0, if 0 < 6 < tt, and
g27ri? _>.oOjif — 7C<6<0.
Using the general as3nnptotic values of 15*3, it is easily proved
by means of the above expressions that
{x) ^ constant x af
when aj->oo along any radius vector other than the negative
real axis which is a singular direction. Similarly, the positive real
axis is a singular direction for These are particular cases
of a more general theorem that the asymptotic properties of the
solutions of a normal equation hold when a? oo in any direction
which is not singular.
15 *55. The Case = ag. When the roots of the character-
istic equation are equal, the difference equation is of normal form
only if ay be a zero of ^o(0* Ir case, writing = ag = a the
differential equation of 15*5 becomes
v' (t) _ AQ(t-b) _ a . (3
v(t) t(t-a) ~
whence v* (t) = (t - a)^ and therefore
% (^) = 2^ j
Put t = az, we have, using 9
u, (a;) =
_ r(a?-a)
~ r(x-a-|-p-hl)TPpl ■
In this case the equation is reducible, and in fact we see that a
second solution is a“ since, by hypothesis,
Aya^+Bya+Cy = 0, 2Aya+By — 0,
r(n-)
CB-a—l
(z- lydz
ar .
15-55] COEFFICIENTS. LAPLACE’S TRANSFORMATION 499
Thus we have found two solutions which are obviously linearly
independent.
When a is not a zero of <j>Q(t) the equation is no longer of
normal form, and if we make Laplace’s substitution, we obtain
^ 2 _ Y
v{t) t t-a {t-af
whence {t- a)^
This yields a solution
(^) = I it - ay eY/(« - «) dt
valid if i^(x~a)>0.
To obtain a second solution, we observe that if
Y = X t^a = r
then JL. =
t-a r
Hence when i->a, if | <^ - 0 | > Jtt and oo if
I ^-0 I <-|tu.
Through a draw a line AB perpendicular to the line joining
a to a+ Y? Kg. 23.
Then 6v/(«--a) q t-^a, provided that t be on the side
of AB remote from u + y- Hraw the contour C as shewn in
Fig. 23, which departs from a on the side remote from a+y
500 LINEAR DIEKEBENCE EQUzVnON WITH RATIONAL [15-55
with Si,ig{t-a) = n + cj} and returns on the same side with
B,Tg{t~- a) = -71+ Then
Uo (x) = f (t - af
Jo
is a second solution of the difference equation, for by the way in
which we have chosen C the integrated part corresponding to
I (x, t) of 15*0 (4) vanishes.
Making the change of variable,
t-a —
and suppressing a constant factor, we have
where L is the contour of Fig. 10, 9*72.
Since the contour C can be made on as small a scale as we please,
we can arrange that
hl{az)\<l.
Expanding the integrand by the binomial theorem, we obtain in
terms of the complementary gamma function, 9*72,
u,(x) = (- (7j r,(- P-S- 1).
s~0
Since
ri(-p) = (-s-i)...(-p-s-i)ri(-p-s-i),
x-oc-l
1
we have
W2(a;) = a“=ri(-p) ^
the series being a Newton’s Series. This case is interesting in that
we have found solutions of an equation which is not of normal form.
(P+1)...([3 + 6‘+1)
15*6. Equations not of Normal Form. Equations whose
coefidcients are polynomials and which are not of normal form may
sometimes be reduced to normal form by a suitable change of
variable. Consider
Pfi {^) Ux+n + Pn-l {X) U^+n-1 + ... + 2Jo(^) = 0,
501
15-6] COEFFICIENTS. LAPLACE’S 'niANSFOKMAl’ICN
and suppose that the (h^grec of is jf, that the degree of
is p + nk, and that the degree of does not exceed p + sh
5 = 1, 2, 1, where Jc is a fixed integer positive or negative.
The equation will be reduced to the nonnal form by the substitution
% =
For, if k be positive, the term becomes, after
division by [r(:/;) P\
[(a; + ,s' - 1 ) . . . {x + ,1 ) xf (a;) ,
whose coefficient is of degree not exceeding
{71 - ,s‘) Ic -f p -i- sk = p “1- n/c,
while the extreme coefficients are of degree p + 7ik. If Jc be
negative, we multiply the equation by
[(a’ + 'n- n-2) ... xY^
and obtain coefficients of degree not exceeding p.
Example.
(ui -f ^2 a; + U3) + (bj^xi- b^) u^+i + % = 0.
Here ;p = 2, /c = - 1. Writing [r(a;)]"'^, we get
foia;“ + a2aj + U3) ^0.4.2+ (:r+ l)(6ia-+52) + = ^5
which is of normal form.
EXAMPLES XV
Discuss the solution of the following difference equations :
1. ((r+ 2) ■“ (7a?H- 3) 'ax’¥iY ~ d.
2. {x ~ 3) %+2 - {4a; - 3) -P (4a;.+ 2) = 0.
3. (4a;+5) %+2 + (12a;+10) w^+t+(9a?+7) = 0.
4. Use the method of 12-72 to shew that the solution of the
complete equation
2 ^ + h) (a? + 5) = f{x),
502 LINEAR DIFFERENCE EQUATION WITH RATIONAL [ex. xv
■where f(x) is a polynomial, can, in general, be made to depend upon
the case in which f{x) is replaced by a constant.
5. With the notation of 15-5, shew that the complete equation
[Ai (a: + 2) + .4 o] “*+2 + (a: + 1) + Bo] + (C^ a: + Cq) = c,
where c is a constant, can be satisfied by taking
M* = X f (< - (< - dt,
J Ol
where X is so chosen that
= c.
6. If be polynomials, shew that in terms of the
operator E Laplace’s equation can be written in the form
If f{x) can be expressed in the form
f (^) ~ |* ^ (0
a
discuss the conditions under which the difference equation will
have a particular solution of the form
= f dt.
' a
In particular, shew that v(t) must satisfy the differential equation
-^{t)v(t)+F{t) = 0.
7. Shew that the equation
{x + a) (a; + b) + ^i) '^x^i f{x)
can be reduced to the normal form by the substitution
Vas = t^a!r(a;+a-l).
Shew how the method of Ex. 6 may be applied to this equation in
the case where f{x) = pxi-q.
EX.XY] COEFFICIENTS. LAPLACE’S TRANSFORMATION 503
8. Apply Laplace’s substitution to the equation
(a: - oc) (a: - ^) M* - [2a:(a; - 1) - S (a: - 1) + a^]
+ (a:-2)(a:-Y-l)w*_2 = 0,
where S = a+p + Y + l-
CHAPTER XVI
EQUATIONS WHOSE COEFEICIENTS ARE EXPRESSIBLE
BY FACTORIAL SERIES
After equations with rational coefficients the next type in order of
simplicity appears to he formed by those equations whose coefficients
can be represented by inverse factorial series. Such equations have
been considered in detail by Norlund who has shewn how to form
series which satisfy certain classes of these equations and has proved
the convergence of the solutions. Norlund’s method consists in the
direct substitution of a series, followed by transformations. It
seems, however, simpler to use an operational method which leads
ultimately to the same series but avoids the transformation of the
terms which is inherent in the method of direct substitution. We
shall begin by establishing the necessary theorems of operation.*
1 6-0. With definitions of Chapter XIV, we have
x' = x-r, Tzu{x) = x' l^u{x),
-1
We shall now prove
Theorem IX.
(Tr + p + o)"^^ p~’- - (TT+fl+ 1) p"^+(7t+a + l){Tc + a+2) p“®- ...
00 ^
S = 1
where the operand is any function of x, and
(j>(ki~a+ly s) = l)(X+(i+2) ... (X+ci^+5).
* L. M. Milne-Thomson, “ On the operational solution of linear difierence
equations whose coefficients are expressible by factorial series,” Proc, Cam-
bridge Phil Soc,, 28 (1932).
504
,6-0] EXPRESSIBLE BY FACTORIAL SERIES 505
Proof. Assume that
(7t + p + a)-i = /o(Tt) p"^+/i(7r) p"H/a(7r) p-» + ... .
Operating with tc + p + a, we have
1 = P /o (Tt) P“^ + [P /i /o P“^] + ■ • •
+ [p Sb{i^) P~’“^ + (tc + a) /.-i M P“'] + • •• ■
Using Theorem I, 14-1, this gives
1 A: (tc - 1 ) + [ /i (tc - 1 ) + (tt + a) /o (re) ] p- H . . .
+ [fs(TC-l) + (-r^ + a)/s.-iM]p~‘ + -" ■
Thus we must have
foa-l) = h
fl (^" 1) + + /o(^) =
/s' (^ “ 1) + fs W “
whence
/oW = l. /i(X) = -(X + a + l), /2(X) = (X+a+l)(X + a + 2),
and generally
/^(X) = (-l)‘'(X+a + l)...{X + a ^.9),
which proves the theorem,
16-01 . Theorem X.
= p-” - (l) (w + a + n) p-”"^
+ (” 2 ^)(Tt + a + n)('n:+a + Ji+l) p-"-^ + .-.
p-” + S ( - 1)* C r ^ P"”"'’
where
^(X+a + w, s) = (X+o+«) ■■■ (X+a + « + s-l),
and the operand is any function of x.
506 EQUATIONS WHOSE COEFFICIENTS ABE [16-01
Proof. The theorem is true when n—1 for it is then the same
as Theorem IX. We therefore proceed by induction. Assume
Theorem X to he true for a positive integer n, that is to say
(1) [(Tu-fp + a + w-l) ... (Tt-fp + a)]"^
A p-" -f 2 ( - ^
V 6 /
By Theorem IX, with a + n for a, we have
[7r+p + a + n]”^= P~^+S + ^ + ^ + P"
8=1
Operatiag with this on (1), we get
(2) [(7r-fp + a + ^^)(7r+p + <3: + ^-l) ... (rc+p + «)]"^
5 = 1 V =0 ^
fn-^s- V- 1
5- V
X ^{7z + a + n + l, v) p-"^“*' ^(TTH-a + ri-, 5- v) p-“«~s+»'
Now, by Theorem I,
(j>{n + a-hn-hl, v) p-^-^ ^(7z + a + 7t, 5- v)
= ^(Tc + a + ^+1, v)^(Tc4-a + w+l + V, s- v) p-^-i-^
^cf>{7z + a^n+l,s) p-w-i-s.
Also
s.(
s-v “,4o ^
/n-i-l + s- 1
N/n+l + s- n
\ n
/ V s J
so that the right-hand side of (2) is
p~n-1^2 ^)^(7U + a + 9^+l, s) P""”"*^"'*^
5=1 \ s /
whence the theorem follows by induction from the case n = 1.
In proving Theorem X we have written in a certain order the
factors in the left-hand member of the statement in the enuncia-
tion. That the order is immaterial depends on the fact that the
same expansion is obtained for [{in:-l-p + a)(7r-l-p + 6)]"^ as for
[ (tt + p + 6) (tt + p H- a) ]"^. This is quite simply proved by assuming
16-0 1] EXPRESSIBLE BY FACTORIAL SERIES 507
an expansion for each as in Theorem IX and then operating
on the first with (7r+p + a)(7r+p + 6) and on the second with
(tt + p + 6) (tt + p 4- a) . These last two operators are equivalent and
we are led to the same functional equations for the coejHicients
in each case.
The application of Theorem X to the theory of difference
equations reposes on the equivalence of tt+p and x-r = x'
regarded as an operator.
Thus with any operand
^ = ^ p-i-(n:+l)p-H(7:+l)(7r + 2)p-3-... ,
x’(x' + l) "" (^^7Hu+p + l) +
+ (2)(’^+2)(’’;+3) p-^-... ,
and so on.
It follows at once that a factorial series of the first kind can be
replaced by an equivalent operator.
Thus with r = 1, x' = x-1,
=:«(, + a-i p-^ + [<*2 - (tt + 2) ] p-^
+ [a3-(i)a2(^+3) + ai(^+2)(Tu+3)]p-3 + ...,
the general term being
(tts - J «S-1 (ti + s) + (^ 2 «s-2 (t' + «) (t' + ® - 1 ) ~ • • •)
vs=0
the product (TC+s)(7r + s--2) ... (tc:4-5~- v-hl) being interpreted as
unity when v = 0.*
* We shall make the corresponding convention throughout the chapter as
the formulae are more readily expressed when it is adopted.
508 EQUATIONS WHOSE COEFFICIENTS ARE [16*1
16*1. First Normal Form. Consider the difference equation
Since
u{x-r)= - A
we can reduce the equation to the form
n n~l
(2) ?«(a:)AM(a!) + ?n-i(a:) A «(a:)+ ... + ?o(®)“(^) = 0.
-1 -1
If a^{x-l){x -2)... (jr-^ + l),
(5 = 1, 2,..., n), and qo{x) = a^,
the equation is of the form which we have called monomial
gv T(x)
(see 14*5) and has solutions of the type ^ ‘ obvious
generalisation of the monomial type is obtained by supposing in
the definition of q^{x) that as is replaced by
,3) <.w = - ■
When X'-^oo the modified coefficient — > and we should expect
such an equation to have solutions which behave asymptotically
in the same way as the solutions of the corresponding monomial
equation.
We shall therefore consider an equation of the form
(4) t^{x){x--l){x-2) ... (x~w+l) A w
-1
1
"t 1 (^) (^ "" 1) (iT — 2) . . . (a? — 92- -f- 2) A W 4” . . . "t" i^Q (oo) = 0,
-1
where j5s(iJ3), 5 = 0, 1, 2, is of the form (3) and where 0.
An equation of this type will be called the first normal form.
Since an=f=0, we can divide (4) by tn{x) and thus obtain an
equation of the same type in which ^^(a;) = 1. We shall therefore
suppose this to have been done.
EXPRESSIBLE BY FACTORIAL SERIES
509
lG-2]
16*2. Operational Solution of an Equation of the First
Normal Form. For simplicity we consider an equation of the
second order, which we write in the form
2
(1) (x-l)(x-2) A ^^-hib‘^q(x))u
Here a and b are constants and p{x), q{x) are factorial series,
(2)
(3)
^ a; ^ x(x+l)^ x{x+l){x-i^2y •’
_^2 I ^ ,
^ 0? '^x(a:+i)'^£r(a;+l)(a;+2) ’
while the indicial equation is
(4) /o(7c) = k{h+l)^-ah+b = 0.
When the roots of this equation are congruent, say
K = K-^p
where p is a positive integer or zero, we take
a(i)=/o(A + l)/o(i+2).../o(7:+y).
The right-hand member of (1) is introduced, as in 14-22, to allow
for this case. Taking a;' = cc~ 1, we have p“* 1 = r(cc) IV{x-\-h),
and we note that the right-hand member of (1) vanishes when
h = and its partial derivate with respect to h vanishes when
k = As in 14*22, we suppose the variation of k to be restricted
to a small region K in the neighbourhood of k^, k^.
Using 16*01 (3), we have
«=1
where
M-^)= ^”^)(7i:+5)...(-n: + s-v+l),
and a similar expression for obtained by writing b^^^ for
Thus, using Theorem II, our equation assumes the operational form
510 EQUATIONS WHOSE COEFFICIENTS ARE [16-2
which can be written
(5) [/o(-Tc)+/i(-'n:)P"^+/2(-'^)p~®+---]'“
= CoOc(A)/o(A;) p-*^l,
where, using Theorem I,
(6) /o(-^) = 'n:(7r~l)-aiT + 6,
/i(-^) = -<ii(7T:)(Ti:+l) + tJ;i(7c),
/2(-7t) = “'^2(^)(^ + 2) + ^2(^)»
and so on.
We can at once obtain a formal solution of this equation by
assuming that
(7) u=^u {x, k) = Cq a (/c) H- q p““^'~^ + . . . ,
where the operand unity is understood.
Equating coefficients, we thus obtain
/o (k) Cq a {k) = /o (X;) Co a (/c), an identity ;
^i/o(^+^) + ^o^(^)/i(^^+l) = 0,
C2/o(^ + 2)4-Ci/i(i+2) + Coa(;fc)/2(^; + 2) = 0,
(^) ^s/o (^ + 5) + (jfc 4- 5) + Cs_2/2 (ifc + 5) + • . •
+ Co(x(*)/3(X; + s) = 0.
As already explained in 14-22, these equations yield determinate
values of c^, Cg, ... in terms of Cq, provided the domain of variation
of k be sufficiently restricted. We have thus found a formal solution
of the non-homogeneous equation (1). Calling this solution u{x, k),
the homogeneous equation
2
(9) (x-l){x-2) ^u-{a+p{x)){x-l) ^u+{b-^q{x))u = 0
-1 “1
has the solutions u {x, k^), u [x, when k^ are incongruent and
the solutions
(10) u{x,\),
when k^ are congruent.
16-3]
EXPRESSIBLE BY FACTORIAL SERIES
16’3. Convergence of the Formal Solution,
found the formal solution of 16-2 (1) in the form
/ 7\ /7\ , r'(ir) r(£c)
(1) u{x, k) - <^0^i^)Y(x+k)^^^r{x+k-rL)'^''^rJx + Jc+2)
which may be written in the form
511
We have
where
u(x, Jc) =
r{x+k) ’
^ (x+k)'^lx+k){x + k+l)^ ‘
Using 2-51, we have
Aw = - [/cv- (a: + A:- 1) A«] ,
A w = [a (A: + 1 ) v - 2/i: (a: + ;b - 1 ) A D
+ {x+k-l){x+k-2) ^v]^.
Substituting in 16-2 (1), we obtain the equation satisfied by v,
namely,
(3) (x + k- l)(a3 + A-2) + 1) A'^
-1 -1
If in the definitions of n and p we now take x' = x+k~l,
then (3) assumes the form
[tc (tc- l)-(2^; + a)7r + fQ(k)]v-p (x) 7iv+[kp{x) + q{x)]v
= CqCc (k) /o (^) .
Since
^(tC” l)-(2^; + a)7c+^;(A; + l) + aifc+6 =/o(--7i: + i),
we have finally
(4:) /o ( - ^ + i) 'y = Co a {k) /o {k)-^[p (a;) {tv ~-k)-q (x) ] t;,
which is the original equation with iz-k written for tt.
512 EQUATIONS WHOSE COKFEICTENTS ARE [16-3
Now, let the factorial series for p{x), q{x) be supposed trans-
formed so that
(5)
2){x) = ^
- a., s I
x'{x' +1) ... (cc' + .s) ’
(6) ij> W+sW = i ■
Then (4) becomes
/o ( - Ti: + fc) w = Co a (/c) /o (fc) + ^ •
Now, siace a:' = 7tr4- p,
as' ...(*' + s)^ N ___ °^sT: + |
T- E S \)(tc + /j+1) - (Tt + s)p-i-^[-a;,7i:+p,.],
f,*=0 As=0
using Theorem X. Thus (4) becomes
(7) /o(-u + fc)t) = Coa(A)/o(/c)+
5 = 0
where
(8) FA-^)
= '^^i-iy~^h\(^^{-K+h+\) ... (TC + .s)[-(Ti:+s+l)(Xft+P;,].
Now we know that (7) has the formal solution (2), which can bo
written in the form
V = Coa(/c) + Ci p“^ + C2 p“^+C3 p-3+ ... .
Substituting this in (7), we get
/o { - 7C + fc) [Co « (Z:) + Cl p-i + . . .] = Co a (ic)/o (7c)
+ [Fo{-n:) p~^ + F^{-tc) p-2+...][Co(x(7c) + Cip-i +...].
Equating the coefficients of the powers of p, we have
<h/o(l + ^) = •^o(l)<^o> =
cM+7c) = F,(2)d,+F^{2)c„
Czfoi^ + k) = ^'2(8)do+J?’i(3)Ci+iro(3)c2,
(9)
— ®s-l'^'o(®) + ®s-2-^l(-^) + ••• +<^0 J'5_] (s).
513
16-3] EXPKESSIBLE BY FACTORIAL SERIES
Now, from (8), we have
Put
= oco + ai+ ... +a;i, ••• + P/i-
Then by Abel’s Identity, 10-07, we have
(10) F.M = ,! g ^;;:J;*)[(v-,-l).4. + £.].
Thus if v> s, a condition always fulfilled by (9), we see that
^’^(v) is a linear form in A/^, with positive coefiBlcients. Hence, if
in the difference equation (3) we replace the factorial series (5) and
(6) for p(x) and hp{x)^-q{x) by majorant factorial series (see
10-091) the numbers corresponding to the J's(v) in the formal solu-
tion will be replaced by numbers which are larger in absolute value
than the numbers -P’s ( v) in (9). We have supposed h to be restricted
to a small region K in the neighbourhood of Aj, We can there-
fore find a number [x ^ 0, independent of k, such that the factorial
series (5), (6) certainly converge if B (x') > [x. We then take as
majorant series (see 10*091) both for p (x) and kp {x) q the function
M. ([X-f* s) ([X -hS-fl) (p.+ S4'5“ 1)
aj'-fx-e x'{x' + l) ,,.{x' -\-s) ’
where s > 0 and Af is a fixed positive number chosen sufficiently
large for the majorant property to hold for all values of h in K,
Now consider the difference equation
2 M V ~\
(11) cx' {x' -1) + +Toj
_i -1 cc— sL -1 -1
where 0 < c < 1 and Yq is a constant whose value will be deter-
mined later. This equation being analogous to (3) has a formal
solution (which we shall presently determine) of the form (2), namely,
= S {x' + l)(x' + 2)...{a^ + s) •
(12)
514 EQUATIONS WHOSE COEFFICIENTS ARE [16-3
The coefficients are given by equations of the same form
as (9), namely,
(13) r.(cs2+l) = y,_iXo(«)+r»-2Xi(s) + - + To)G-i(s),
but the numbers Xai'^), which correspond to are now all
positive, as is easily seen from the form analogous to (10). More-
over, we have shewn that
(14) ''>«•
Now, since 0 < c < 1, we can find a positive integer n, such that
(15) cs® + l<|/o(*+s)!>
for all values of h in K. Also, by successive applications of the
recurrence relations (13), we have ys = ®'sYo where nig is a
positive number, while in a similar way from (9) we obtain
Os = '{'«(*).
where ^g{h) is a rational function of k. It follows that if m be
the smallest of the numbers 1, mj, m„, and if
(j;>max[lCoa(fc)l, l'l'nW|]>
we shall have
(16) T,>lc,l, s = l,2,...,n, yo>lCoa(*)|,
provided that y, be so chosen that myo > i}'- If fhis condition
be satisfied, it follows from (13), (14), (15), (16) that, interpreting
Cji-n as do,
Y«+i> S l/o(^+»» + l)l > l<=n+il>
from (9). We prove in the same way that > | c„+2 | and so on,
and thus we have y, > | c, | for v = 1, 2, ... , n, nH-1, ... . It
follows from this inequality that, when the series (12) converges,
the series (2) also converges. To establish the convergence of (12)
we substitute this series in (11), which gives in operational form
OO 00
(c,.>+i)gY.p- = ;;5:jqr=-.(-’'+l)ST.p-+T..
515
16-3] EXPBESSIBLE BY FACTORIAL SERIES
whence, using Theorem IV, we obtain
00
(7c+p-fi.-e)2 (cs2 + l)Y,p-*
= (s+1) Ysp“® + (7t+p-ti.-s)Yo-
s=0
Using Theorem IV again, we obtain
J; [ - (3+ IX + e) (c 32+ 1) y, P-+ (c s^+1) Y, P-+1]
«==i
= ilf 2(s + l)Y,p->.
S = 0
Equating the coefficients, we obtain
(c-i-1)yi=: Myo,
Tm[^(^ + ^)^ + l]-(^+f^+e;)(c32 + l) Ys = M(s4*l)
Thus
Ys+i _ (g+|x+s)(cs^ + l) + M(34-l) _ (3 - Ij) (s - 12) {s - Iq)
Ys c(5+l)^+l ($-nbj){s-m2)
say, where
(17) ^2 "t" ^3 “ "" "t" ^2 ^
Thus if Ts, T5+1 be consecutive terms of (12), we have
^5+1 _ (g — li) {s — 12) {s — Iq)
Ts (s - Mj) {s -m^{x' + S+1)
— I + + I Q
$ \5V *
and therefore by Weierstrass’ criterion (see 9*8), the series (12) is
absolutely convergent if
JJ(a?' + Zi+i2 + l3-mi-m2 + l)> 1,
which gives, using (17),
E(x') > p.+ e~2.
Since x' — x+k- \ we see that (12) converges in the half plane
determined by
(18) I2(a;+X;)>(x-1.
516
EQUATIONS WHOSE COEFFICIENTS ARE [I6.3
We liave therefore proved the coavergence of (12) and therefore
of (2) and consequently of (1) in the above half plane. Since the
convergence of a factorial series is uniform in a half plane interior to
the half plane of convergence we have established the existence as
analytic functions of the solutions u(x, u{x, of the difference
equation 16*2 (9), at least in half planes defined by (18).
Moreover, since pi is independent of h, the convergence is uniform
with respect to ifc, so that we have established the existence of the
solutions 16-2 (10) when the indicial equation presents congruent
roots. The linear independence of these solutions follows at once,
as in 14*3, from their asymptotic behaviour when a; co in the half
plane of convergence.
16*4. Example of Solution. Consider the equation
(1) (<*2 + |) {x- l)(a;-2)AM- (% + !) (»- 1) Am+ («o + ”) u = 0.
This equation has rational coefficients. We shall, however, here
regard it as an equation with factorial (in this case finite) series for
coefficients. The equation is of the first normal form, if 0.
The indicial equation is
foik) = a2k(k-\-l) + a;^k-\-aQ = a2{k-hy){k-k^.
Take x'=. x- I, then the operational form of the equation is
■p +
~ ^0 /o(^) P~^*
When the roots of the indicial equation are congruent, say
^2 ~ ^ ^ 0, we take
^0 = ^0 /o(*+l)/o(*+2) .../o(/fc+p).
When the roots are incongruent, we take do =
Expanding by Theorem IX, we have
617
J0.4] expressible by factorial series
whence, using Theorem I,
1 _
[62(7C + S)(7C + S-l)-&i(7t + s) + &o] P“®| M - <^o/oWp ^
which becomes
(2) [/o(-'^)+/i("'^) + + = dofom 9"^’
where
f^{-Tt) = (-1)‘'-^(tc + 2)...(7U+v)/i(-t:- v + 1),
(3) L+i(lc + s+l) = (fc + S-1) ... (fc+s- v + l)/i{* + s- v + 1)
= {h + s-l)Mk + s).
Now substitute
u = dQP~'^'\-Cip ^ ^ + ^2? ^""^+***
in (2). Equating the coefficients we get
c^Mh+l)+d^A(}c+l) = 0,
(4) c, /o {k+s) + c,_i A{k+s) + ... + df,A{k+s) = 0,
(5) c,^^fo{k+s + l) + c,A(k+s + l) + e,_^A{k + s + l) + -
+ ^0
Using (3), we get
c,+ifQ(k+s+l) + c,A{k+s + l)
+ (/c + 5-1)[C5^1 /i(/c + 5) + ...+^^o /o(^ + ^)]
whence, using (4),
Cs+i /o(fc + 5 4-l)+Os[/i(^J + 5 + l)-(ifc + 5-- l)/o(^+'S)] — 0.
Put
kfo{k + l)- fi{k+l) = <*2 (A; - y (* - k) - *3)>
AW =^b.,{k-mi-l)(k-m2-l)-
Then
^s+1 _ ^2 (^ “■ ^1 + S-l){h-l2 + S-l){k-‘lz±£ _
Os ” a^ik— /ci+s+l)(^—
do ~ a2(fc+l“--fei)(^+l-"fc2)
0.
0,
and
518 EQUATIONS WHOSE COEFFICIENTS ARE [16-4
Thus using the factorial notation
(k~li + s-lY^Hlc-l2+s-lY‘'>{k-l3+s-lY^'>IJc-m^) {h-m^h^g^
Thus (2) has the solution
u{x, h) = Aq +^2 ...
+ 1
where
(yi;„ ^^+54. i)(s+i)(ij^ 7.2+5 + (xTk + s)
and the solutions of (1) are u(x, ki), u(x, k^) when /c^, k^ are in-
congruent and u{x, k^), du{x, / dk^ when k^, are congruent.
16*5. The Second Normal Form. An equation which can
be expressed in the form
Tn(x)x{x-^1) ... (aj+w-l)
n-l
+ Tn^i{x)x{x + l),..{x-{-n-2) /S,u+ ... + To(cc)'a = 0,
where
s = 0, 1, 2,
and where 0, is said to be of the second normal form.
The operational method of solving such an equation is exactly the
same as that already explained except that the oj)erators iz, p are
replaced by p^. The basic theorem for these operators is
Theorem XI.
(-7Ui-a+pi)-i= pf i+K+a+l) pj-2
+ (tU3l + C3&+ 1) (tCj + <X+ 2) p£'^+ ... ,
where the operand is any function of x.
The proof of this theorem, which is analogous to Theorem IX,
offers no difficulty and is left to the reader.
16-5] EXPRESSIBLE BY PAOTOBIAL SERIES 519
An equation of the second normal form has solutions of the form
and partial derivates of this with respect to when the indicial
equation presents multiple or congruent roots.
The region of convergence is R{x) < max(X+w, R{l + h)),
where X is the smallest abscissa of convergence of the coeJBScients
of the given equation and k is the root of the indicial equation
whose real part is smallest.
The proof of these statements follows exactly the same lines as
that for the first normal form.
The solutions have the same asymptotic forms as those of an
equation of the first normal type and form a fundamental system.
16*6. Note on the Normal Forms. Consider the equa-
tion
(1)
p^{x) A A «+?»o(®) u = Q.
-1
-1
If polynomials of descending degree, the
equation can be reduced to the first normal ty^pe, for we can write
the equation in the form
and since the degree of the numerators of
p^{x){x--2) Pf^(x)(x--l)(x-2)
p^ix) ’ p^{x)
does not exceed the degree of the corresponding denominators these
rational functions can be expressed in factorial series of the type
necessary for the equation to be normal. Now, with v{x+2) =u(x)j
(1) can be written in the form
2
b>2 (^) +Pii^)+Po (^)] A [Pi W + 2po(a7)] A ^+3^0 (a?) = 0,
and the coeflicients are again of descending degree, so that this
equation can be reduced to the second normal form.
520 EQUATIONS WHOSE COEEFICIENTS ARE [16-6
Thus we see that an equation like (1) in which the coefficients are
polynomials of descending degree can be expressed in both normal
forms. We can thus get two fundamental sets of solutions according
to the tjq)e to which we reduce the equation. These correspond to
the two canonical sets of solutions discussed in Chapter XV.
More generally an equation of the first normal form
2
l)(ci?~*2) A A u-\-tQ{x)u = 0
-1 -1
can be reduced also to the second normal form, provided that the
factorial series which represent t^{x), t-y(x), t^ix) be holomorphic
in the whole domain of the point infinity.
If a given equation cannot be reduced to either normal form it
may still be possible to obtain a certain number of solutions by the
operational method. These will not of course form a fundamental
set, for the normal forms constitute the only type in which the
solutions are all represented by the class of factorial series already
obtained.
Example.
(2-^)a;(a;+l)A«-(l-^)a;AM + w = 0.
The equation is expressed in the second normal form.
The indicial equation is, taking a?' = x,
fo (^) ~ = (27c -}- 1 ) (/c -f- 1 ) = 0,
whence = - 1.
These are incongruent, so we write
- TCi+ 1 + pi-
)%(%-!) -(l-
__1
■7ti+pi,
TCjL-h 1
w = 0.
Expanding by Theorem XI, we get
r/o(“%) X) ... (tCi+s-2) 1)
-(TCi+l)...(TCi + S-l)pf'7Tjj”jw = 0,
[/o ( - %) - S (tCi - 1) (71:1 + 1) K + 2) . . . + s) pf 'J U = 0.
ie-6] EXPRESSIBLE BY FACTORIAL SERIES 521
Put
M = Co pr*+cipi'*~^+c2pr*'“‘+...,
then
Co/oW=<^. Cj/o(Jb+l)-Co(^: + 2)yS: = 0,
Csfo(Jc + s) — Cj_j^ (7c + s + 1) (^ + s — 1)
+ C3_2(7/4' s + 1)(7:+5” 1)(7/-I'S'-2) “ ...
+ (-l)*Co(7:+s + l)a+s-l)(/c+s-2)...it = 0,
Cj+i /o ^ “t ^ ) ~ Cs (^ + s + 2) (7: + s)
4-Csji(7c + s + 2)(7:+s)(7:+s- 1)- ...
+ ( - l)“+ico(7:+s+2)(/fc+s)(7; + s- 1) ...k = 0.
The last two equations give
C5/o('^+c) c,+i/o(ii:+^+I) „
jt + s + 1 ■^(i!: + s + 2)(7: + 5) ’
whence we obtain
2m- (7‘^+c)(^ + c)
Cs k+s+i^
( -1)» (7: + s)(yfc + 5-l)... (ib+l)(7:+g)(7: + s-I)... (7:+l)
(7; + s + #)(7; + s+l) ... {k+ .])
Thus we have
u{x; -|) =
u{x; -1) =
k
r (x+^)
T{x)
r(cc+i)
V{x)
1-
r+
1
4:(x-iy2^(x-i)(x
- 1 = X- 1.
f)
EXAMPLES XVI
Solve the following difference equations :
1. (a:-l)(a:-2) A (k- 1) A^w+(cio + ^) u = 0.
2. (a:-l)(a;-2) Au-(2 + -)(a:-l) Am+(2+J)u = 0.
522 COEFFICIENTS EXPBESSIBLE BY FACTORIAL SERIES [ex. xvi
3. (a:- l)(a;- 2) (l +^) A ““ A «+ 12« = 0.
4. (®-l)(a;-2)(4+^^)A«
5. (a:+l)(a;+2)(l-^)Aw
_(i_^)(.4-i)A«+(i-^> = o.
6. a:(a;+l)(4-^)A“-(8-^)»A« + 9w = 0.
7. a:(a:+l)(l-— )AM-(5-^):^At^+(8--|i)M = 0.
8. a!(a:+l)(4-^)AM-(8+^)a;Aw + 9« = 0.
9. Establish for the operators tc^, Pi the theorem corresponding
to Theorem X.
THE CALCULUS OF
CHAPTER XVII
THE THEOREMS OE POINCARfi AND PERRON
In this chapter we discuss certain theorems on the asymptotic
behaviour of solutions of linear difference equations. The theorem
of Poincar6 * marks the beginning of modem methods of research in
the theory of linear difference equations. The failure of the theorem
in certain cases leads us to discuss the theorem of Perron. The
proof here given is Perron’s and is based on the properties of sum
equations.!
It has been considered advisable to reproduce here the whole of
Perron’s paper both on account of the elegance of the method
employed and also to give an insight into the theory of sum equa-
tions which have an interest of their own apart from the particular
application in view.
17*0. The Linear Equation with Constant Coefficients.
Consider the equation
(1) u{x+Z)-{ix.+ ^+y)u{x+2)
+ (ap+ Py+ ya) w(a:+ 1) -apY «(») = 0,
where a, §, y are constants. The roots of the characteristic equation
are a, y. If we put
(2) aL = rie'\ =
we have
* H. Poincar4, American Journal of Math,., 7 (1885), p. 213.
f 0. Perron, “ tfber Summengleiohungen und Poincar^sche Differenzen-
gleichungen,” Math. Annalen, 84 (1921), p. 1.
523
524 THE THEOREMS OE POINCARE AND PERRON [17 0
We propose to investigate the value of
uix-^n + l)
lim z ' r >
u{x^n)
(3)
where w is a positive integer.
Case I. kl > I PI > I yI-
The general solution of (1) is
M (cc) = c?! (a;) a® + {x) p® + (x) y®,
where Tx^ix), m^{x) are arbitrary periodics, We dismiss
once for all the trivial solution u{x) = 0, which corresponds to
the case in which these arbitrary periodics are all identically zero
Let us choose an initial value of x, say a:^, for which roi(a;o)=tO.
Then
u(x^+n + 1) _ ^i(^o) tt®<'+"+^+ ga(^o) P®"^”^^ + (x^) y-
u{Xo+n)
«*«+” + 072 (Xo) P®'>+”+ 073 (Xo) y
Since
(4)
C7i(Xo) + ,a72(Xo) 0 H-C73(Xo)0
a7i (Xo) + 072 (a^o) (^) ” + mg (x^) (|)
, are less than unity, we have
w(Xo+w+l)
u(Xo + «.) “ '*■
Zo+n-f-i
Similarly, if C7i(x3) = 0 while t!73(Xo) 0, we have
m(Xo + «. + !) _ ^2(^0) P‘^"'*~"'*'^+ WgCXa) y»n-”+i
u{Xg + n) ~ xng (Xp) p»o+n+ 1 + (x^®»+“+i ’
so that
(6) = p.
^—>00 u{xQ-{-n)
Finally, if ru^ixQ) 0 while vifiix^) = 0, = 0, we have
(6)
91 ->00 '^(ajQ+w-)
17-0]
THE THEOREMS OE POINCARfi AHD PERRON 525
Thus, if u{x) be any solution of (1), which is not zero, we have
proved that
n-^co +
is equal to one of the roots of the characteristic equation.
Case li. a == p, |a| > | yj.
In this case
u {x) = (a:) a® + (x) xoL^-{-rn^ {x) y®.
Suppose that when x = Xq, 0. Then
u{xQ + n+l)
u(xQ + n)
so that
a(a:o + n+l)
_Xo+n+l "^^Xft+n+1
^Y^iCo+W+l
(xg + n)
_Xo+n XQ + n\cx./
^u{x^+n+l)
n->ao
Similarly, if = 0, the value of the limit is
again oc, and if ^^^{Xq) = 0, ^2,i‘^o) = 0, 'w^{Xq) 0, the limit exists
and is equal to y. If |a|>lp|, p = y, a similar conclusion is
ched.
Case 111. a = p = y.
In this case
u (x) = w-^ (x) a® + ^2 (^) {^)
and we can easily prove as in Case II, that the limit (3) exists and
is equal to oc.
Case IV. lal = lpl, e^<^, |al>ly|.
In this case
u (x) = uTi (x) r® -1- uf2 (x) (x) y®.
If ixri(Xo) =/= 0, t2y2(^o) ^
i^o+n+l)_ &i(Xo) e»(^o+”+» ^ + Wj, (Xp) e*(^o+"+i) ^
u(xg+n) - ®i{a;o)e»(*«+»)®+®s(a;o)e‘(*»+»)^
526 THE THEOHEMS OE POINCAR^: AND PERRON [17-0
Since 6^”*^ do not tend to deifinite Kmits when n -» oo , we
see that the limit (3) does not exist. For particular solutions the
limit may exist; for example, if rEr2(a;o) = 0, mj(xQ)=f^0, the
limit is a, while for ^2(^0) ^ limit is j3, and
for = 0, t272{i*^o) = ^ T*
Thus in this case we can state that the limit (3) does not always
exist.
The cases |al>|p|, 1|3|=|y|, (f>=hi^ and | a | = | (3 1 = | y |,
0 ^ ^ are similar to the last and do not require separate
discussion.
The method of reasoning evidently applies to a homogeneous
equation with constant coeJQSlcients of any order, and we can state
the following general theorem.
Theorem. Given a homogeneous linear difference equation with
constant coefficients, let u{x) he any solution such that u{xQ)=f^0,
Then, if n be a positive integer,
n-¥-<ao uiXiQ-t-n^
exists and is equal to a root of the characteristic equation, whether
these roots be distinct or not, provided that those roots which are
distinct have distinct moduli.
If the characteristic equation have two or mare distinct roots with
the same modulus, the above limit does not in general exist, but
particular solutions can always be found for which the limit exists
and is equal to a given root of the characteristic equation.
17*1. Poincare’s Theorem. Poincar6 has generalised this
theorem for equations whose coefficients tend to constant values for
large values of the variable. For simplicity we consider an equation
of the third order. The particular initial value Xq which figures in
the above theorem may by a displacement of the origin be taken as
unity. We therefore consider the equation
(1) -a(^+3)^-[a4■cc(?^)]^^(n^-2)
+ [6 + y(n)]w(n+l) + [c+2J(n)]u(n) = 0,
17*1] THE THEOREMS OF POINCARfi AND PERRON 527
where n is a positive integral variable and a, 6, c are constants.
If when n-^oo ,
(2) lima;(7^) = 0, limy(n) = 0, lim2:(n) = 0,
we shall call an equation of the form (1) a difference equation of
Poincare’s tjpQ,
When n is large, the difference equation (1) approximates to the
form
(3) u{n’^Z) + au{n+2)-\-bu(n + l) + cu{n) = 0,
which we may call the associated equation with constant coefficients.
With these definitions we may state the following theorem :
Poincare’s Theorem. If u{n) be any solution of a homo-
geneous linear difference equation whose coefficients tend to constant
values, when n + oo , then
li-m
n-^ao
u(n+l)
u(n)
exists, and is eqical to one of the zeros of the characteristic function
of the associated difference equation with constant coefficients, 'provided
that the moduli of the zeros of the characteristic function be distinct.
We prove the theorem for equation (1), which is of the third
order. The characteristic function of the associated equation (3) is
(4) f{t) = t^^at^+bt-\‘C^
and we can suppose that
(5) I 1 > I “2 I > K I
since, by hypothesis, the moduli are distinct. It also follows that
f (*i)> f (*2). S' (“3) different from zero. Now put
(6) u (n) = Pi (n) + Pa (w) +P3 (w),
u(n+l) = aiPi(«) + a2P2(w) + a3P3(m),
u(n + 2) = a?pi(n)+o(|p2(n) + «|p3{«).
These equations are compatible since
1
Oi
1
*2
1
«3
= (oq- 02) («2- Os) K- *1) i= 0-
528 THE THEOREMS OF POI'NCARH AND PERRON [17-1
Multiply equations, (6), by -{x^ + a^), 1 respectively
and add. We then obtain
02 Ogwl^)- (oj + ag) u{n+l) + u{n + 2)
= [a? -“1(02 + “s) + “2 03] Ti (»»)=/' (“i) Pi («) •
Writing {??-+ 1) for n, we get
(7) a2a3^^(rz- + l)- (ag + ocg) u{n-{-2)-{-u{n-\-?>) = /'(ai)Pi(^+ 1).
Substitute the value of u{n-i-3) from (1) and observe that
(Xi + Og+ag = -a, aia2+a2a3+a3ai = b, cciCi2^z — -c.
We then have
r K) ?i (^ + 1) = [u {n + 2) - (02 + 03) + 1 ) + 02 03 u (n) ]
-x{n) u{n-\-2) - yin) u{n-\-l)-z{n) u(n),
whence, using (6) and (7),
/'(a,)j,,(n+l)
=«i/' K) Pi W - -X^i(w) Pi(w) - X2(n) pg (n) - Xg (/i) pg (m),
where
Xs(n) = (x,^x{n)-^asy{n) + z{n), s = 1, 2, 3,
so that, from (2), Xs{n) — > 0 when n -> co .
Thus we have the three equations
(8) Pi{n+1) =aij)i(n)-^i(w)yi(«)-r]i(re)2J2(n)-ri(«)pg(w.),
^2 {?^ + 1) = 052 P2 - ^2 {^) Pi (n) - rii (n) p.^ (to) - 1(2 (to) p.j (to),
Jig (to + 1) = Og Jig (to) - ^g (to) Pj (to) - TJg (to) Pg («) - Cg (to) Jig (to),
where = ^^(n) -^/'(al), so that the coefficients
7)5 (^), Zs (^) when n 00 ,
Since 1 ] > | ocg j > | 03 |, we can choOse a positive number [3
such that
(9)
l + P
<1,
I«31 + P
<1,
l3J± | <
I «i I - P
it being sufficient to take 2^ less than the smaller of
17-1] THE THEOREMS OF POmCAR^] AND PERRON 529
Since the coefficients ^,(n) in (8) tend to zero,
we can find a positive integer such that the absolute value of
each of these coefficients is less than provided that w > Wq .
We exclude the trivial case u(n) s:- 0, from and after some fixed
value of n. It then follows, from (6), that Pi{n), p^in), p^{n)
do not vanish identically. Now take a fixed integer and
consider the sequence of functions
\PiiN)\, \P,(N)1 \p,{N)\.
As we proceed from left to right in this sequence we must, at a
definite stage, first come to a function whose value is at least as
great as the value of any of its successors in the sequence.
Let I Pi{N) I be the function defined in this way.
If we change N into N + 1 we shall shew that the suffix i cannot
increase. It will then follow that i will tend to a limiting value when
N -^00 ; for i cannot increase and has one of the values 1, 2, 3.
The possible distinct types of inequality between the functions of
the sequence are
(A) \Px{N)\>\p,{N)\, \p,{N)\>\PziN)\, i-=l;
(B) |p,(F)|>|ft(iV)|, \p,(N)\^\p,(N)\, i = 2-,
(C) \P,{N)\>\P^{N)\, b3(A)|>|2^,(iV)|, i = 3.
Since |a + 6|^|a| + |6|, |a-6|^|a|-|6|, we have in case
(A), ftom (8),
|pi(A+l)|>(ai||^,i(A)|
lp,(A+l)|<|a,||p,(A)|
+ -^P{lPi(A)| + b2W| + b3(A)|}<[|«3| + p]|j,i(A)|,
lj)3(iV + l)K|a3| IPsiN)]
+i^{\piim\+\p,m+\pzm}<i\^z\+mpim-
Thus, by division, since pi{N) 0, we have, using (9),
p^{N + l) ■
< °^I + P
<1,
which shews that case (A) when once established for sufficiently
large values of n will persist for all greater values of n.
530 THE THEOBEMS OP POINCAEfi AND PEBRON [17-1
In case (B) we have from (8), in the same way as above,
|P2(iV+l)l>[|«2|-P]l?2W|.
lP3(^+l)l<[l“3l + P]|?aWI>
and thus, using (9),
MN+1) ^KI-P ’
so that I j32('^ + 1) I > 1 1) I, and therefore
if 1 2’2 (•^+ 1) 1 > I ft (-^ + 1) U (®) persisted, while
if 1 ft(-ZV^+l) 1 ^ lft(-^ + l) l> (B) has become a case of (A).
Evidently (C) either persists or becomes (B) or (A).
Thus we have proved that the suffix i cannot increase, so that for
sufficiently large n, i remains constant.
Suppose, for example, i = 2. We now prove that
limn(i) = o, lim»y' = 0.
l->ooj^2(^) ri’-^coPzi^)
(10)
For suppose, if possible, that
where l> 0. Then, given s > 0, we have, for sufficiently large
values of n.
ftW
ftW
< Z + e.
Suppose that N be chosen large enough for this to be the case,
then, from (8) and (B), we have
\P,{N + 1)\>\^\\p^{N)\-^\P,{N)\,
\P2{N + 1)\K\c^\\P,(N)\ + ^\P,{N)\,
and thus, by division,
i«iiift(^)i-piftW
{|«2l + P}lft(^)
Pi{N + l)
P^{N + 1)
< <Z+s,
and thus
17-1] THE THEOREMS OE POINCARfi AND PERRON 531
Now, from the property of the upper limit (10-08), we have
PAN)
PAN)
for infinitely many values of N.
>l-s
Thus we have
which gives
<(^ + s){ia2l + P} + P>
kil + KI + P
which is impossible, since s and p are arbitrarily small, and
I I - I «2 I > 0.
Thus we must have I = 0, which proves the first part of (10).
The second part is proved in the same way. Then, from (6), we
have
and thus
lim
n-^co
u{n)
p^)
= 1,
lim
n— >-00
u{n+l)
PAn)
=
lim
n— >-00
^(^+1) ■■■ ..
u(n)
which proves Poincare’s theorem in the case i = 2. The cases i = 1,
r= 3 present no new features. Thus Poincare’s theorem is proved
for the third order equation.
The method of proof for the equation of general order follows
exactly the same lines, the essential point being the proof that the
suffix i cannot increase.
Poincare’s theorem shews that
hm ^ -■ — ~
is equal to one of the roots of the characteristic equation. A more
general theorem has been proved by Perron,* namely :
If the coefficient of u{x) in the difference equation of order n be
not zero, for x= 0,1,2, ... , and the other hypotheses be fulfilled, then
' 0. Perron, Journal f, rein. u. angew. Math. 136 (1909), 17-37.
THE THEOREMS OF POlNCARJi! AND PERRON
532
[17*1
the equation possesses n fundamental solutions Ui{x)^ such
that
a!->oo ^i{^)
i = 1, 2, ... , n,
where a^- is a root of the characteristic equation, and x oo by positive
integral increments.
When the conditions of the enunciation of Poincare’s theorem are
not all fulfilled, that is, when the characteristic equation presents
two or more roots of the same modulus, the matter becomes very
complicated, and it may be shewn by examples that the theorem
may even fail completely. We shall discuss another theorem due
to Perron which frees us of these complications.
17*2. Continued Fraction Solution of the Second Order
Equation. We first establish a certain identity due to Thiele.
Let
(1) ^
(2) ^
Then
and thus we obtain
l.55±1Z
i?i±?
^s+2 “
-^3+3’
■ ^s+1
fiL
~ 'fin
^ X
^s-l-2
s = 1, 2, ... , n-3,
1.
«--2
and hence
^s+l
— s — 1, 2, 3, — 3,
Proceeding in this
(3)
%
way we obtain the identity
1
^n--4
17*2]
THE THEOREMS OF POINCAR3i3 AND PERRON
533
Now consider the Poincare difference equation
u{x+2)+p {x)u{X’{‘l)i-q (x) u (x) = 0,
where
lim p {x) = lim q (x) = a^,
where a; oo by positive integral increments.
The characteristic equation is
t^+a^t+a2 = 0.
We shall suppose the roots a, ^ of this equation to be of unequal
modulus and that | a | > | (3 ] .
Now let Ui{x)j U2^{x) be a fundamental system of solutions. We
then obtain from the difference equation
, ^ ^1 (a; + 2) Ma {x) -u^{x+ 2) Mj (a;)
' U^{x)U2{X+l)-U2{x)u.^{x+l)’
, _ Ui{x+\)u^{x+2)-u^{x+l)u^(x+2)
^ u^{x)u^{x-{-l)-u^{x)u-^{x-^l)
If in (1) we take
._.Ml(g + g-2)
u^{x-^s-2y
we have
(A\ g(a;+g~l)
^ ~ p(x+s-2)p(x+s-l) ‘
Writing n + 2 for n, we have, after reduction,
^ ~ u^(x-hn)u2(x)-U2(x-i-n)u^(x) p(x-l)'
Substituting in (3), we have, by means of (4), the identity
Uj^(x+n) U2{x+ 1) - U2(x-hn) %(ag+ 1)
^ ' Ui(x + n}u2(x)-U2(x+n)Uj^(x}
p {x) - ^ —
p(ic+l)-
q{x-j-n-2)
p{x+n~-2)'
534 THE THEOREMS OP POINCAR^J AND PERRON [17-2
The right-hand member of this identity depends only on the
coefficients of the difference equation and is therefore independent
of the particular fundamental system chosen. Let us choose our
fundamental solutions so that
.“r. - p'
which is possible by Perron’s theorem, given at the end of 17-1.
Then
lim + ^ u^{x+n)
from which it follows that
Hence, dividing the numerator and denominator of (5) by (a: + w)
and then letting « oo , we obtain
«2]£±1)
_ -£(X)
y(,)- !(S+1)
^(03+1)-
g(a!+2)
33(a:+2)-
and u^{x) is obtained as a solution of an eq[uation of the first order.
In a similar manner, by writing -t-2 for x in the difference
equation, we can prove that a second solution is given by
u^jx)
«3(a:+l)
-1
pix-l)-
q{x-l)
p(x-2)-
g(a;-2)
p(x-d)-
17" 3. Sum Equations. By the name sum equations we
understand a system of infinitely many equations in infinitely many
17-3] THE THEOREMS OF POINCARE AN'E PERRON 535
unknowns, such that in the {\x+ l)th equation the first unknowns
are absent.*
Thus we can write such a system in the form
%0 + + ^2 + %3 % + ••• =^03
0 ^2"^%, 2 %+••• =
^2, 0 ^2 ^2, 1 + . . . = ^2,
or, more briefly,
00
(1) S >■ [A = 0, 1, 2
v—O
We assume that
(2) K>0, 0<%<1,
(3) lim sup y Icju, K 1.
(Ji — >• 00
We then seek solutions for which
(4) lim sup J/ ] ajJ < 1.
For such solutions the series (1) are absolutely convergent. Now
let
/(2)=
be an arbitrary power series, which for 1 2; j 1 is holomorphic
and different from zero, so that the reciprocal
m
v=0
is likewise holomorphic for 1 2; | ^ 1 and different from zero. It
follows that the radii of convergence are both greater than unity,
so that I
(5) lim sup y 1 Tv 1 < 1, lim sup V 1 Tv I < h
* I have translated the German “ Summengleichungen” by “sum equation”.
The equations form a semi-reduced or semi-normal linear system. The idea
of these equations is due to J. Horn, Journal /. rein. u. angew. Math. 140,
(1911).
t K. Knopp, Infinite Series, p. 155.
536 . THE THEOREMS OF POINCAR^l AND PERRON
Moreover, from the definition of the coefficients,
X X
(6)
Y y' = V Y ' T = / ^ ^
Z^Tx-kTk [0forX>0.
If then we put
[iKi
(7)
^/a+X, v-xT\
X-0
(8) ^ ^M+x Tx
x=o
we obtain without difficulty
(9) K'>0, 0<^y<L
(10) limsup
/It ”>00
From (7), we have
V V V — K
^ J ^ll + K, V - K Y«' ^ J ^ j ^/X + #C-}-X, V - K - X Tx Y«
K = 0 <C = 0 x = o
from (6), Again, from (8), since by (3) and (5) the double series is
absolutely and therefore unconditionally convergent, we liave
QO T
^/it+KY« ~ ^^+/c+x Yx Yff
/c = 0 ic~0 X—O
= S S Vt-xYx~KY« = ^
x=0 »C = 0
from (6). Hence we have proved that
= s
x=o
2 «m+x,v-xYx-«T« =
K=0
(11)
^ y + V - K Y'
IC = 0
00
(12)
<+K yk = <
K = 0
If now we form the sum equations
(1^) [jL = 0, 1, 2,
y=-0
17.3] THE THEOREMS OF POINCAR^] AND PERRON 537
these are equivalent to (1) in so far as every solution of (1) (which
satisfies (4) ) is also a solution of (13) and conversely. To see this,
it is sufficient to shew that (13) follows from (1).
The converse is then obtained by interchanging the letters with
primes and those without. In (1) put p-f X for pt, multiply by yx,
and sum with respect to X. Then
00 00 00
2 2 ^M+x, V ^x+M+v Tx = 2 ~
X=0 v=0 X=0
and since by (2), (4), (5) the double senes converges absolutely, by
interchanging the members, we have
00 V 00
2) 2/ ^/^ + X, V “X Yx + V 2»’
V =0 X = 0 v=‘0
from (7), so that (13) follows from (1).
17-4. Homogeneous Sum Equations with Constant
Coefficients.
Theorem I. Let the coefficients of the homogeneous sum
equations
00
^ j — 0, 1, 2, ...
be such that =/= 0 and such that the function
F(z) ^a,z^
is holomorphic and has w ( ^ 0) zeros {multiple zeros being counted
according to their multiplicity) in the region | 2: | ^ g^. Then the sum
equations have exactly n linearly independent solutions for which
lim sup V I iT J ^
V — > 00
These solutions are
where is a zero of F (z) of order
538 THE THEOREMS OF POINCARfi AND PERRON [17.4
That the above values are solutions is easily verified, for the
statement amounts to proving that
= 0, 5 = 0, 1, 2, ... , mx-1,
when 2: = px .
Clearly we can take j = 1, since the substitution
q~^,
brings us to this case.
Since the series is now convergent for | 2: | < 1, we have
[a, Z>0,
so that condition 17*3(2) is fulfilled = «>). Let
P{z) = z«+gj^z«-T-+...+g„ = g„_^ z"
V.--0
have the same zeros as F{z) for | z j ^ 1. Then the function
P(z)
F{z)
= /(«) = s YvZ’'
y^O
is holomorphic and has no zeros if | z J < 1 and therefore fulfils the
hypothesis of 17-3. The sum equations can therefore lie trans-
formed and, hy 17-3(7),
V
But multiplying the former equation by F(z), wo liave
so that
v=0
V 9n—v3 — 0> 1) 2, , Uf
V = 0, V > n.
The transformed sum equations are therefore
^ j Qn-^-y ^fL+y —
V =a0
= 0, [X = 0, 1, 2, ... ,
or
17-5] the theorems OF POINCARfi AND PERRON 539
wliicli are linear difference equations witli constant coefEcients
whose solutions are just those given in the theorem. The point of
the theorem lies not so much in the fact that the given values are
solutions as that there are no further independent solutions.
17*5. A Second Transformation, Returning to the hypo-
theses of 17*3, let us put
(1) p. == 0, 1, 2, .. . 5
thus obtaining the equations
^ J i^V "b y)
s*=0
and assume that for all p
(2) % + by,Q=f=0,
so that in the (p4-l)th equation the unknown actually occurs.
Moreover, let
(3) 0<^<1,
(4) linl ky, = 0.
00
Finally, let the function
(5) Fiz)=f,a,z^
be holomorphic for 1 2; | < 1, so that, if necessary replacing ^ by a
greater number which is still less than unity, in addition to (3),
we have also
(6) layt<6S^^ 6>0.
Then the hypotheses of 17*3 are fulfilled. If ^ 0 be the number
of zeros of F(z) in 1 2^ K 1 (counted according to multiplicity), we
let
(7) P{Z) =
be the polynomial with just those zeros. Put
(8) |.-y = f{z) = J] Yv a*'
540 THE THEOREMS OP POINCARlS AND PERRON [17.5
Then f{z) is holomorphic for 1 2 1 < 1 and has no zeros in this
region. We can therefore use the transformation of 17-3,
whereby (1) becomes
00
(9) ^ j p) 1, 2, ,
wtere, analogously to 17*3 (7), (8), we have
(10) <= S «v-xTx,
x=o
(11) ba V = ^ + X Tx J
x=o
(12) 1 X Tx?
x=o
and in particular from (2),
(13) ao + &M,o = («o + ^a.,o)Yo=^"^'
Multiplying (8) by F{z), we get
S 9n^v 2:’' = 2 2^' 2 Tv
V=a0 V=0 V=bO
SO that
al == 2 <^v-x Tx = ffn-vj V = 0, 1, 2, , n,
x=o
and aj = 0, for v> n. The transformed sum equations therefore
take the form
00
(14) 5^1 •^^4-71— 1 ■P "h ^ M ^jjL+v P* “ h, 1, 2, ...j
and, by (13),
(1®) 9n+bil ,o=f=0.
For 6;,„ from (3), (4) and 17-3 (6), we have
(16) \K,y\<K^''', 0 <»■'<!,
(17) lim kl^ = 0.
/« -ri" CJO
We also obtain
(18)
lim sup 'y|c;i<l.
17-5] THE THEOBEMS OF POINCARE AND PERRON 541
In solving the system, (14), it is clearly sufficient to satisfy
these equations for [x > M, where M can be as large as we like.
When we have done this the missing unknowns
can, on account of (15), be found from the equations (14) by
putting successively p = M - 1, M - 2, , 0.
We shall now transform (14) when M, leaving the precise
determination of M till later. Put then
(19)
A majorant function for this series is clearly
so that we have
(20)
(^~1)-
?i4- V - 1\ 1
V
V
Also, from (19), multiplying by P{z), we get
(21)
Zj ^
and hence Sq = 1. Now, in (14), put successively
11 = M, M+1, M+2, ..., II
and we have
00
*^=0
CO
+724-1 + S^l^iwr+n+ ••• +^72^ilf+l = <^i»f+l“ 2 ^M+I+VJ
V=0
^/ji+7i "t 9l ®ju.+n-l “t ••• '^9n V
v=0
Multiply these equations in order by ^ So
add the results. Then, from the identity (21), the numbers
^iU+n? ^M+n+l? ••* j
542 THE THEOBEMS OF POINCAB.fi AND PERBON [17-5
disappear and, since Sq = !> obtain
n-1
(22)
„=o x=»o
It. - M 00 M - M
X = 0 v=0 X = 0
where the 8 with negative suffixes which may appear on the left
are to be replaced by zero.
The system (22) for pi = M, lf + 1, M + 2, , is clearly the full
equivalent of the system (14) for the same values of pt. The
condition 17 *3 (4) is here unnecessary, and wc shall therefore allow
solutions of (14) and (22), for which this condition is not fulfilled.
17*6. General Solution of Sum Equations. Let be any
number in the interval
(1) i<?:<i/^'.
Then from 17*5 (17), if M be sufficiently large, wc have
(2) v^M,
an inequality which still holds if ^ be rejdaccd by another
number sufficiently near to !^. We can therefore determine two
numbers ^2? such that, in the interval
i<^i<^<^2<i/^y,
we have
(2a) A;<4(i-yg(^3L-“i)’s
(26) Aj;<i-(i-yg(i:2-i)« v>m.
We now prove that, if the number ^ and the index ill be chosen
to- satisfy (1) and (2), then the sum equations 17*5 (14), when
are arbitrarily assigned, have exactly one solution such that
limsup ly
p— >00
From 17-5, we See that it is suflS.cient to consider the system
17-5 (22).
17-0] THE THEOKEMS OF POINCARE AND PERRON 543
In the first place it is easy to shew that there is at most one such
solution.
For, if possible, let be two such solutions. Then their
difference satisfies the homogeneous sum equations :
(^) ~ V -X,
x=o
and
(4) Zm = Zm+X = = 0.
Also, since
lim sup 1 cc, 1 < ‘C < ^27 lini sup 17 1 2/. ! < ^ < ^2^
we can find a number G such that
(5)
We here take C to be the smallest number for which this holds,
which is possible since the aggregate of all such numbers clearly
includes their lower limit. Then, from (3), using 17-5 (16) (20),
the relation (5) gives
00 f*.-M
„=o x=o
X=0 ^
x=o ^ ^
where, ia the last line but one, we have used (26).
Taken in conjunction with (4), this states that
in other words, that in (5) we can replace (7 by |C/. Since G was
already chosen as small as possible, we must have G ='0 and
therefore a:,, = y,, for (jl ^ M . Thus we have proved that there
is at most one solution of the prescribed Mnd.
To prove that there is actually one solution, denote the prescribed
initial values by
••• J ~
544 THE THEOBEMS 01? POINCAR^! AND PERRON [17-6
and let us seek to solve the system 17-5 (22) by successive approxi-
mation, putting
(6) p > M 4- n,
(7) M < p < M -t- w - 1,
iJi- M oo n- M
(8) s c;_xSx- s S
^ ’ x=0 v-O X-0
n-l V
-s s i/n — v+X iir — X
>- = 0 x=o
We first shew that the successively formed series converge in that
(9) \x>M,
where C is independent of p and s.
From 17-5 (18), (20), we see that
\c'A<K,X,t,
n- 1
v+x ilf “ X ^M+v
x=o
where K^, independent of p. Now, on account of (7), no
proof of (9) is needed for M ^ \x ^ M+n-l. Also for s = 0 no
proof is necessary. If then (9) be true for a certain value of s, we
have from (8),
*i’ <'S
x=o ^ ^
y=0 X=0 \ A /
and if we approximate by the same method as that just used in
discussing (5), but using (2a) instead of (26), we get
+ 1 c cr*-" -h
If then we take
we have proved (9) by induction.
17*6]
THE THEOREMS OF POINCAR]!: AND PERRON
545
From (8), it follows that
00 (i-M
(10) = - ^ T 6' S IxjW 1
^ M+n zL/ M - X, V - X
v=0 X = 0
We now prove that
(11)
By (7) no proof is needed, for M + Also (11) is
true for 5 = 1, by (9). If (11) be true for a certain value of 5,
we have, from (10),
I i< 2 s"
again using the method employed in treating (5). Hence (11)
follows by induction.
From (11) we infer the existence of the limit
(12) lim = X ,
and in fact
From this we get
-X I < -
00 ft- iKl
--Yn+n
once more using the method of approximation adopted for (5).
Hence
00 M oo/jl-M
lim E ^K-x.A^.+^-x^
>-00 v-sO x*=0 v=0 x=0
546
THE THEOREMS OF POINCARi: AND PERRON [17-U
and hence, when s -> oo , equation (8) become equation 17-5 (22).
The solutions obtained by the successive approximations therefore
all satisfy 17-5 (22) and therefore 17-5 (14). They also satisfy the
postulated condition, for from (13) and (9),
(14) limsup
jui—>00
so that the proposition is proved. But from this proposition, in
conjunction with (14), wo can draw a further conclusion, namely,
that if a solution of 17-5 (14) satisfy the condition
lim sup s/ 1 I 57 ,
fX—^-QO
then the sign of equality never occurs.
Now 11 is any number in the interval (1) and hence 11 cati Ije
taken to differ from unity by an arbitrarily small (juantity, so that
we can replace (14) by the sharper inequality
lim sup 1 ijx^ I ^ 1.
— > 00
For solutions wMcli satisfy this condition the sum equations
17*5 (14) are equivalent to the sum equations given at the beginning
of 17-5, namely,
^ y "b ]k)
so that we have solved these also.
If these equations be homogeneous so that all the and theridbre
all the cl are zero, then there are n linearly independent solutions,
which can be fixed with, say, the initial values
1 ^M+n-l,l 1- ^
I ^M,n ®
The general solution has then the form
0
0
1 J
0 0
THE THEOREMS OF POINCARi^ AND PERRON
547
17-G]
where the are arbitrary constants. In the non-homogeneous
equation the difference of two solutions is clearly a solution of the
homogeneous equation. If then x^= be a particular solution
of the non-homogeneous equation, the general solution is
n
X = 1
We may sum up all these results in the following theorem.
Theorem II. Let the coefficients of the sum equations
00
y; K + = Cm> 0, 1, 2, ...
satisfy the conditions ; ag + 6^^ „ ^ 0, [ji = 0, 1, 2, ... ,
lim = 0, lim sup ^ | | < 1-
jU.— >00
Let the f unction
he holomorjdiic for | s | < 1. If n{^0) be the number of zeros of
F{z) in this region, counted according to their multiplicity, then the
general solution of the sum equations which satisfy the condition
lim 1 03,^. K 1
H-^00
contains exactly n arbitrary constants and has the form
n
0 +
If M be a large enough index, there is one, and only one, such
solution for which the n unknowns
^Jkf> •**» ^M+n — X
have prescribed values. For n = 0 there is exactly one solution
and no arbitrary constants.
548 THE THEOREMS OF POINCARfi AND PERRON [17-7
17'7. Difference Equations of Poincare’s Type. Con-
sider the following difference equation of the rth order,
(1) w([x + r) + a,,_,_iw([x + r-l)+...-l-a,,,iM((x-+l)-Fa,,_oM((i,) = 0,
where the independent variable p, takes the values 0, 1, 2, 3, ....
This equation is of Poincare’s type (see 17-1) if the limits
lim V = 0, 1, 2, ..., r- 1,
IX —> 00
all exist. We now prove
Perron’s Theorem. Let q^, q^, ••• ? ?cr ho the distinct moduli
of the roots of the characteristic equation
and let he the number of roots whose modulus is q^^, muUi/ple roots
being counted according to their multiplicity, so that
Then, provided that a^,o be different from zero for all values of
the difference equation (1) has a fundamental system of solutions,
which fall into a classes, such that, for the solutions of the 'kth class
and their linear combinations,
limsup 1(/|m((^)1 =
>• 00
The number of solutions oj the hth class is l^.
Let the numbers qx be arranged in ascending order of magnitude
0<ii<iz<9:3< ■■■ <<!<,■
Let p be an arbitrary positive number and let
(2)
x a - a = V = 0 1 2 r - 1
OCj, — , y tA-j, — 5 y Vj X J *J, . . . J / X,
F F
a, = 1 = ^1, 0 = V>
f’ p^
(3) u{\i.)=p>^x^, p. = 0, 1, 2, ....
Then the difference equation (1) is equivalent to
^ J i^v "h v) ^fX+V
17-7] THE THEOREMS OF POINCARE AND PERRON 549
Let us regard this new equation as a system of sum equations
whose coeflEicients clearly satisfy the conditions of Theorem II, since
they vanish for v > r. The number of zeros of the function
F{z) =
for 1 2: 1 ^ 1 depends on the choice of the positive number p. If
we choose p smaller than (provided 0), tliere are no zeros
in I 2; I 1 and hence no solution, other than zero, for which
limsup <1,
fX—hCO
that is to say, for which
lim Slip C/ 1 w((jL) I ^ p <qi.
ft — y 00
If we choose for p a number between q^ and q^, there are zeros
and therefore solutions, such that
limsup 1 I < 1,
ft— >00
that is to say, for which
lim sup y I w((ji) 1 ?2-
fi—)- 00
Since p can be taken arbitrarily near to q^, we have for these
solutions
limsup jy l^(p-) i = ?i.
ft — 00
If now we choose p between g'2 and q^, there are Z1 + Z2 zeros
and hence solutions for which
lim sup ^\u{[i)\^p <qQ.
ft— >aD
The solutions already found are of course included among
these ; for the others, since p can be taken arbitrarily near to g2,
we have
limsup y |'?^(pL)l = ^2-
ft— > 00
Proceeding in this way the theorem is proved.
650
THE THEOREMS OF POIHCAB^ AND PERRON [ex. xvn
EXAMPLES XVII
1. In tie case of the equation
t(n + 2)-(l+y|”)MW = 0,
shew that
m(w+1)
lim '• , r-'
„^o m(»)
does not exist for any solution at all.
[Perron.]
2. In the equation
M (re + 2) - [2 + (n)] w (w + 1) + [1+ 2>o («)]«(«) = 0,
where
PoW" n~>oo,
and for sufficiently large n, j)i(w)>0, Pi{n)-po{'n)^0, shew that
lim
w-»-oo
u{n+l)
u(n)
= 1,
for every solution which is not constantly zero tor large values
ofn. ■ [Perron.]
3. In the equation
u(n + 2)+2)^{n)u{n+l)+pQ{n)u{n^ = 0,
where p^{n)->0 when n->oo, and where
lim
n-^<x3
23i(w-l)Pi(n)
= a,
where a is not a real number > prove that for every solution
which does not constantly vanish for large n,
Um = 0. [Perron.]
n-^oo u(n)
4. Shew that the limit given in Poincare's theorem does not
exist in the case of the equation
u(n + 2)-
(n + 2) + 2(-l)^
(ti "V 2y^ (fi + 3)
u(n) = 0.
Hx.xvn] THE THEOREMS OF POINCAR15 AND PERRON 551
5. Let a be a number whose modulus is greater than that of
every root of the characteristic equation of a difference equation
of Poincare’s type. Prove that
lim
n oo
- A
a" “ ’
where u (n) is any solution of the equation. [Poincare.]
*6. Let u{n) be any solution of a homogeneous linear difference
equation of order r, with constant coefficients, and let
m(w) . . . u{n+m-l)
D{nr, n)
•It. (n + m - 1 ) u{n \ m) ... n {n + 2»4 - 2)
where m£^r and w is a positive integer.
Then if D(m, n) 0, prove that
lim D{m, n+l)l D{m. n)
exists, and is equal to the continued product of m zeros of the
characteristic function, provided that those zeros which are distinct
have distinct moduli. [Aitken.]
*7. If the difference equation of Ex. 6 be replaced by an
equation of Poincare’s type, shew that the corresponding result
(which is a generalisation of Poincare’s theorem) still holds, pro-
vided that no two zeros of the characteristic function of the
associated difference equation with constant coefficients have the
same modulus. [Aitken.]
♦I have to thank Dr. A. C. Aitken for communicating these elegant
generalisations.
INDEX
The references are to pages.
Abel’s identities, 276.
Abscissa of convergence -See Con-
vergence.
Adams-Bashforth process, 183.
Aitken, A. C., 98, 103, 109.
Aitken’s generalisation of Poincare’s
theorem, 551.
interpolation by iteration, 76.
quadratic interpolation, 78.
quadrature formulae, 199.
Alternant, 9.
Ascending continued fractions, 330.
Barnes, E. W., 484.
Bendixson, 272.
Bernoulli, 124.
Bernoulli’s function, periodic, 187,
210, 326.
numbers, 37, 127, 300.
as determinants, 138.
Generating functions of, 134.
of the first order, 137.
Polynomials, 126 et seq.
Bessei’s interpolation formula, 68.
modified formula, 71.
Beta Function, 262, 288, 484, 486.
Complementary, 267.
Double loop integral for, 266.
Expansion in Newton’s series of
the reciprocal of, 316.
Single loop integral for, 265.
Bohr, 272,
Boole, 49, 343, 387, 434, 436, 444.
Briggs, 47, 101.
Broggi, XJ., 401.
Burnside, W., 198, 415.
Burnside’s formula for double inte-
gration, 198.
Carlson, F., 310.
Oasorati, 354.
Cauchy, 9, 19.
Cauchy’s residue theorem, 11,
*221.
Chappell, E., 74.
Chrystal, G., 8.
Collins, 59.
Complementary argument theorems,
128, 145, 230, 237, 249, 251.
Complex variable, 220.
Confluent divided difference, 13.
Confluent interpolation polynomial,
16.
Confluent reciprocal difference, 117.
Constant in Norlund’s definition of
log r{x), 252.
Continued fractions, 108, 120, 330, 378,
532.
Convergence, Abscissa of, 276, 309.
Landau’s theorem on, 279.
order of singularity and, 292.
Weierstrass’ criterion for, 260.
du Bois-Raymond’s test for, 274.
Cotes’ formulae, 168.
Cubic, Approximate root of, 4.
Davies, W. B., 98.
Dodekind, 276, 277.
Derivates, First order, 103, 154.
from Bessel’s formula, 161.
from Stirling’s formula, 159.
Functional, 369.
Markoff’s formula for, 157.
of higher oi’der, 155,
Reciprocal, 118.
Dienes, P., 226, 292.
r>3
554
INDEX
Difference, Central, 22.
in terms of derivates, 37, 162.
notation, 20, 22.
operators, 20, 22.
quotients, 23, 24.
quotients in terms of derivates, 37.
quotients of zero, 36, 134.
See also Differences, and Divided
differences.
Difference equations,* 322.
Adjoint, 374.
Asymptotic forms of solutions, 457,
487.
Binomial, 465.
Boole’s iterative method for, 343.
operational method, 392.
symbolic method for, 387.
Bronwin’s method for, 475.
Canonical forms of, 443.
systems of solution, 482.
systems in factorial series, 485.
Casorati’s theorem and determin-
ant, 354, 357, 373, 385, 488.
Characteristic equation, 479.
Characteristic function, 384.
Clairaut’s form, 344.
Complementary solution, 389.
Complete, 328, 374, 388, 460.
Complete primitive of, 322, 390.
Conditions for exact, 337.
Convergence of solution, 459, 511.
Exact, 334.
Exceptional cases of, 451.
Existence of solutions of, 352.
Formal solution of, in series, 445.
Fundamental systems of solutions
of, 353.
General theorems on, 357, 360, 443,
526, 527, 531, 548.
Genesis of, 322.
Haldane’s method of solution, 341.
Homogeneous, 324, 346, 351, 384.
Hypergeometric solutions of, 494.
IndQcial equation of, 446, 509.
Linear independence of solutions
of, 360.
Linear, of the first order, 324, 328,
329.
Method of variation of parameters
for, 375.
Milne-Thomson’s method, 410.
Miscellaneous forms of, 347. •
Monomial, 461.
Multiple solutions of, 370.
Multipliers of, 339, 372.
Non-linear, 341, 346, 420.
Normal forms of, 478, 508, 518.
519.
Not in normal form, 500.
Partial, 423, 475.
Partial fraction series for, 490, 495.
Particular solution of, 390, 401
404, 406, 407.
Reducible, 366, 493.
Reduction of the order of, 367.
Relation between fundamental sys-
tems, 359.
Relations between canonical sys-
tems, 496.
Resolvable into first order equa-
tions, 426.
Riccati’s form of, 346.
Simultaneous, 420.
Singular points of, 352.
Solution by continued fractions,
330, 378, 532.
differencing, 344.
Gamma functions, 327.
Laplace’s method, 427, 476.
operators, 392, 410, 413, 509.
undetermined coefficients, 403.
Solution in Newton’s series, 448.
Special forms of, 472.
Symbolic highest common factor
of, 361.
lowest common multiple of, 363.
Transformations of, 467, 478.
with coefficients e.xpressihlo in
Factorial series, 504.
constant coefficients, 320, 384,
423, 523.
polynomial coefficients, 377.
rational coefficients, 434, 478.
Differences, Ascending, 22.
Backward, 22, 59.
Descending, 22.
for subdivided interval, 87.
Forward, 22, 56.
in terms of derivates, 162.
Numerical applications of, 87.
of a numericaf table, 88.
Reciprocal, 104.
as determinants, 110.
confluent, 117.
of a quotient, 112.
Properties of, 114.
See also Difference and Divided
differences.
INDHX
555
Diflereiitiiil e<|wati<)n, mnnerical hoIii-
ium of, 183.
Piffcrentiai <‘C|uation of Fiujhwian
ty|H3, 478.
Diflt^rentiation, Kmm^riaiU 103, 154,
I)ivi<ku:l diilcrencea, 1.
as coiitoor integrals, 11.
as dotertninants, 0.
as delinite, integrals, 10.
confluent, 13.
for equidistant argumtsnts, 50.
in terms of functional values, 7.
of 7.
du Bois-lteyinond, 273, 274.
Eldertori, W. Palin, 40.
Elliptic integral, complete, 70, 85.
Error 'Pest, vSteflensc'ms, 02.
Elder, 124, 252, 257, 270, 311.
Elder- Maciiiurin formula, 187, 210.
Euler's constant, 245.
polynomials and numlxirs, 143.
of the lirat order, 147.
transformation cf series, 311.
Everett’s interpolation formula, 72.
Existence of the iirincipal solution or
sum, 209.
Expansion of circular functions, 138.
in powers of x, 133.
in f.ictoriais, 133.
Exponential function as a continued
fraction, 121.
sum of, 231.
Expression for F(x | - e>), 238.
Factorial expressions, 25.
of the form x(^), 25, 42.
of the form x<"“^), 25, 44.
Integral of, 131.
moments, 41.
series, 271.
Associated, 272.
Convergence of, 273.
.Finite difference and sum of, 300.
for canonical solutions, 485.
Inverse—^'ee Inverse factorial
scries.
Newton’s— Newton’s series,
region of absolute convergence,
276.
region of convergence, 275.
Theorems on, 272, 275.
^jgurate numbers, 52.
’inite summation, 191.
Korsytli, A. E., 482.
Fourier series, 218, 247, 326,
Frobenius, 434.
Function, Bernoulli’s periodic, 187,
210.
Beta — See Beta function.
Gamma — See Gamma function.
Holomorphic, 221.
Hyporgcometric, 261, 264.
Incomplete Gamma, 331, 407.
Integral, 226, 230.
Meromorphic, 221.
Prym’s, 332.
Psi— /See Psi function.
Functional derivates, 369.
l<'unctions with only one singular
point, sum of, 232.
Gamma function, 249.
Asymptotic properties, 254.
Complementary, 258, 500.
Complementary argument theorem
for, 251.
Duplication formula for, 257.
Generalised, The, 255.
Hankel’s integral for, 259.
Incomplete, 331, 407.
Infinite products in terms of, 251.
fnte^al for, 257.
Multiplication theorem for, 257.
Residues of, 252.
Bchlomilch’s infinite product for,
250.
Gauss, 19, 23, 257.
Gauss’ backward formula, 65.
forward formula, 65, 73.
interpolation formula, 63.
method of integration, 173.
Generating function of Bernoulli’s
polynomials, 127.
Bernoulli’s numbers, 134.
Euler’s polynomials, 143.
Euler’s numbers, 147.
inverse factorial series, 290.
Newton’s series, 312.
Genocchi, 18,
Goursat, E., 478.
Gregory, J., 47, 59.
Gregory-Newton formula, 59.
Gregory’s formula, 191.
theorem, 33.
Gudermann, 54.
Hadamard, 292.
INDEX
Haldane, J. B. S., 341.
Halving the tabular interval, 84.
Hardy’s formula, 171.
Hayashi, K., 108.
Hermite, 10, 124.
Herschel’s theorem, 32.
Heymann’s theorem, 357.
Heymann, W., 357, 488.
Hobson, E. W„ 176, 220.
Holomorphic, 221.
Horn, J., 535.
Hughes, H. K., 294.
Hyperbolic functions, 101, 232.
Hypergeometric function, 261.
Definite integral for, 264.
Expansion in Newton’s series, 316.
for ir = l, 261.
Hypergeometric series, 260.
Hypergeometric solutions of differ-
ence equations, 494.
Indefinite summation, 301.
Integral, Contour, 11, 221, 404.
Integral function, 226, 230.
Integration by Lagrange’s inter-
polation formula, 164.
Numerical, 162, See also Quadra-
ture.
Interpolation, 55.
Aitken’s quadratic process, 78.
by iteration, 76.
formula, Bessel’s, 68.
Central difference, 85.
Everett’s, 72.
Gauss’, 6k
Lagrange’s, 8, 15, 75.
Newton’s, 2, 11, 13, 57, 59.
Steffensen’s, 74.
Stirling’s, 67, 155.
Thiele’s, 106.
to halves, 84.
inverse — See Inverse interpolation,
polynomials, 14.
without differences, 75.
Inverse factorial series, 284.
Addition and multiplication of, 295.
An asymptotic formula for, 298.
Differentiation of, 297.
Integration of, 299.
Majorant, 283.
Poles of, 287.
Theorems on, 287, 295.
Transformations of, 293, 294.
Uniform convergence of, 284.
Inverse interpolation, 95.
by divided differences, 96.
by iterated linear interpolation,
97.
by reversal of series, 100.
by successive approximation, 99.
Jacobian elliptic functions, 78, 97.
Zeta function, 80.
Jeffreys, H., 412.
Knopp, K., 138, 147, 260, 261, 274,
277, 304, 312, 535.
Lagrange, 375.
Lagrange’s interpoiation formula, 8,
15, 75, 164.
Laguerre’s polynomial, 321.
Landau, E., 272, 279.
Landau’s theorem, 279.
Laplace, 427, 478.
Laplace’s difference equation, 491.
formula, 181, 193.
Application of, 183.
integral, 288, 314, 407, 478.
Legendre, 257.
Leibniz’ theorem, 156, 211.
Analogue of, 34.
Levy, H., 184.
Lidstono, G. J., 100.
Limes superior, 277.
Linear independence, 353, 300.
Log r(a;), 86, 249.
Logr(a?-{-l) as a definite integral,
257.
Lubbock’s summation formula, 193.
Maclaurin’s theorem, Secondarv form
of, 60.
Majorant properties, Theorems on,
310.
Markoff’s formula, 157, 162, 192,
Matrix notation, 108, 379.
Mean value theorem, 163.
Mechanical quadrature aVcc (hiadra-
turo.
Meromorphic, 221.
Milne-Thomson, L. M., 38, 70, 72,
78, 80, 85, 94, 07, 99, 101, 109,
124, 330, 334, 378, 410, 434,
504.
Mittag-Leffler, 332, 490.
Moments, 40.
Factorial, 41 .
INDKX
ihfi
Moltiplutatioii Umnvtm, Ml, 2o,%,
24i\ t!u.
Nevilles K. H.. ^
Kovilltt’K pn,H*t*.s,s of itondioTt, jsl.
ISfowtoiu -T1, -1.14.
Newtoii’n iiitrrpolati'tiii fitnnula, 2,
li, 13. r.T, ,VJ.
• auries, *U)2.
ikmViTi^t'nrt' !ihsriss:i of, 30P.
Kxp;insi«iii in, 3iN), 315, 31^1.
linifonu fonvt/rgoiU’n of, 3o2.
Nioiatni, 11. P.. ISI.
Niekni, N.. 272, 233.
Xoriund. N.K., 124, 2(H), 201 , 202, 203,
241, 272, 2H4, 200, 204, 302,
311.483, 4H7, 480. 304.
NtirlunO’rt «l«4j!Ul-ion of hnr 240.
oprratt)r A, 23.
Proper! i<\s of, 30.
theorem, 4O0.
Null KerieH, 304,
OpiTatioiiH with P"^ on a ^iven
funetion X, 412.
unity, 411.
Operator />, 23.
A, 23.
y, 31.
1“', 31, 32.
P h 37.
'Fheorem on, 30.
r, 430, 437.
TTi, 430.
p, 4,34.
/,,, 430.
Operators, x\j)plieations of the
op(*,rator P"^ to
dilTerencc ccpiations, 413.
dynamics, 416.
energy, 417.
geometry, 415.
linear o.scillaior, 418,
probability, 415,
General theonmis on r, ttj, />, />i,
439, 467, 504, 518.
BelatJons between /), A, E*", 33.
OrdtT of an integral funetion, 226.
Order of singularity, 202.
Partial fraction series, 245, 330,
332, 490, 405.
Partial .snininalitm, 41. 206, 243.
Pas(‘al, 170.
Pt*ri<Hb<* fjiiadion, arbitra.rv, or peri-
od ie, 324.
Pemm, ()., 107, 121, 523, 531, 550.
Perron’.s tluasrein, 54S.
PljaK<% 58.
Pinehti'ie aiui Amaltli, .361, 360.
Plana’s formula, 257.
Pt)ineare. 217, 244, 523, 550,
Pcjineare's theorem. 526.
P<»l<‘. 221 .
Polynomial, Laguerre’s, 321.
Sum of, 208.
IVdynomials, /•), 126.
142.
</., 124.
Heniouili’s, 126, 204, 213, 338.
Ctunplementary arguimmt theo-
rem for, 128.
in interval (0, 1), 141.
of su<*ee.ssiv<* orders, 120.
td the lir.st order, 136.
Propertit^s of, 127.
Ihdation to fa.et.ori{d.s, 120.
I>ool(‘’s t-h<‘onnn for, 140.
DilTenuiee (pu>tients of, 28.
KultT-Maelanrin theorem for, 130.
Kuler\s, 143.
Compkunent-ary argument theo-
rem for, 145.
of sticeessive orders, 145.
of thc! first order, M6.
Prop(‘riieK of, 144.
Kx})ansiou in factorials, 27.
Intorpolat'ion, 14.
Con fluent, 16.
Legendre’s, 176.
Prym, F. E., 332.
Prym’s functions, 332.
Psi function, 241, 268.
Asymptotic behaviour for large
values, 244.
Oomplomontary argument theorem
for, 249.
Differentiation of, 241.
Duplication theorem for, 247.
Expansiem in Newton's series, 315.
Fourier series for, 247.
Gauss’ integral for, 247.
Integration of, 242, 256,
Multiplication theorem for, 246.
Partial fraction development, 245.
Poisson’s integral for, 248.
INDEX
558
Quadrature formulae nvolving differ-
ences, 180.
central difference, 184.
of closed type, 170.
of open type, 172, 199.
Remainder term in Newton's for-
mula, 5, 61,
terms, 166, 167. ^ec aiso individual
formulae.
Residue, 221.
theorem, 221.
Application of, 222.
Rolle’s theorem, 4, 156, 175,
Schlomilch, 250.
Sequence, Upper limit of, 277.
Series, Euler’s transformation of, 311.
Factorial — See Factorial series.
Fourier, 218, 247, 326.
Generating function of factorial,
290, 312.
Inverse factorial — See Inverse fac-
torial series.
Newton’s — See Newton’s series.
Null, 304.
Stirling’s, 253.
Sheppard, W, F., 22, 55.
Sign for symbolic equivalence, 32.
Simpson’s rule, 171, 197.
Singular point, or singularity, 221.
Singularity, Order of, 292.
Staudt’s theorem, 153.
Steffensen, J. F., 62, 85, 166.
Steffensen’s intcrpola.tion formula,
74.
Stirling, 272.
Stirling’s formula, 254.
interpolation, 67, 155.
series for log r(a; + h)t 253.
Subtabulation, 91.
Sum, Asymptotic behaviour
for large values, 214.
for small values, 216.
complex variable, 222.
DijBEerentiation of, 213.
Existence of, 209.
Fourier series for, 218.
of exponential function, 231.
of a polynomial, 208.
of squares of first n natural num-
bers, 43.
or principal solution, 20L
Properties of, 204.
Sum (upiation.s, 534.
General solutiion of, 542.
H omogentM ) uh, 53 7 .
Theonuns on, 537, 547.
Transformations of. 5;I9.
8ummabl(i function, 203.
Summation, 200.
Indefinit(^ 301.
of finite^ st'riea, 42.
of series of polyiiomiaLs, 4,‘h 4(5.
of serie.s of rat-iona! functions, 45.
Partial, 41, 243.
analogy with iutt'gralion by
parts, 42.
Repeated, 208, 243.
Sylvester, 420.
Symbolic highest common factor,
361.
lowest common multiple^ 363.
Taylor, 13, 33, 50, 119.
Theta fumdaon, 72.
Thiel(^ T. N., 104.
Thi<4(‘’s identity, 532.
interpolation formula., 106.
theorem, 119.
Thomp.son, A. J., 74, SS.
Thrce-eigliths rule, 199.
Trapezoidal rule, 170,
TschebyschetT, P., 177.
Tschebyscdieif’s formula, 177.
Turnbull, H. W., lOl, 109.
Uni(jue devtOopmc'nt, *105.
7’heor(un of, 2SS.
Upper limit, 277.
Value of N 137.
1
Vanderm()nd(\ t), 134,
van Orstrand, G. K., 63, 68.
Wallis’ tluiorom, 268.
Waring’s formula, 291,
Weddle’s formula, 172,
WeiorstrasH, 274, 290.
Weierstrass’ definition of t.h<3 ( Jamma
funotion, 250.
criterion for convcjrgenoe of series,
260.
Whittaker and Robinson, 98.
Whittaker and Watson, 11, 217, 222,
245, 251, 252, 258, 259, 260,
272, 277.
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